text:
\begin{document}
\title{On Hypergraph Lagrangians and Frankl-F\"uredi's Conjecture}
\author{
Hui Lei
\thanks{Center for Combinatorics and LPMC, Nankai University, Tianjin 300071, China,
   ({\tt hlei@mail.nankai.edu.cn}). This author was supported in part by National Natural Science Foundation of China (No. 11771221).}
\and
Linyuan Lu
\thanks{University of South Carolina, Columbia, SC 29208,
({\tt lu@math.sc.edu}). This author was supported in part by NSF
grant DMS 1600811.}
}
\maketitle

\begin{abstract}
Frankl and F\"uredi conjectured in 1989 that the maximum Lagrangian, denoted by $\lambda_r(m)$, among all $r$-uniform hypergraphs of fixed size $m$ is achieved by the minimum hypergraph $C_{r,m}$ under the colexicographic order.
We say $m$ in {\em principal domain} if there exists an integer $t$ such that
${t-1\choose r}\leq m\leq {t\choose r}-{t-2\choose r-2}$. If $m$ is in the principal domain,
then Frankl-F\"uredi's conjecture has a very simple expression:
$$\lambda_r(m)=\frac{1}{(t-1)^r}{t-1\choose r}.$$
Many previous results are focusing on $r=3$. For $r\geq 4$,
Tyomkyn in 2017 proved that Frankl-F\"{u}redi's conjecture holds whenever ${t-1\choose r} \leq m \leq
{t\choose r} -{t-2\choose r-2}- \delta_rt^{r-2}$ for a constant $\delta_r>0$. In this paper, we improve
Tyomkyn's result by showing Frankl-F\"{u}redi's conjecture holds whenever ${t-1\choose r} \leq m \leq
{t\choose r} -{t-2\choose r-2}- \delta_r't^{r-\frac{7}{3}}$ for a constant $\delta_r'>0$.
\end{abstract}

\section{Introduction}
The Lagrangians of hypergraphs are closely related to the Tur\'an densities in the extremal hypergraph theory.
Given an $r$-uniform hypergraph $H$ on a vertex set $[n]:=\{1,2,\ldots,n\}$,  the Lagrangian of $H$, denoted by $\lambda(H)$,  is defined to  be
$$\lambda(H)=\max_{{\bf x}\in \mathbb{R}^n_+: \|{\bf x}\|_1=1}\!\!\!
\sum_{\{i_1,i_2,\ldots, i_r\}\in E(H)} x_{i_1}x_{i_2}\cdots x_{i_r},$$
where the maximum is taken over on a simplex $\{x\in \mathbb{R}^n \colon x_1,\ldots, x_n\geq 0,
\mbox{ and } \sum_{i=1}^n x_i=1\}$. A maximum point ${\vec x}_0$ is called an {\em optimal
legal weighting} and the set of its nonzero coordinates in ${\vec x}_0$ is called a {\em support}
of $H$ (see section 2 for details.)
 One can show that $r! \cdot \lambda(H)$ is the supremum of
edge densities among all hypergraphs which are blow-ups of $H$. It has important applications in the Tur\'an theory.

The concept of Lagrangians of graphs was introduced by Motzkin and Straus \cite{MS1965} in 1965, who proved that $\lambda(H)=\frac{1}{2}\left(1-\frac{1}{\omega(H)}\right)$,
where $\omega(H)$ is the clique number of a graph $H$. Their theorem implies Tur\'an's theorem.

 Let $\lambda_r(m)$ be the maximum of Lagrangians among all $r$-uniform hypergraphs with $m$ edges.
 Then Motzkin-Straus' result implies $\lambda_2(m)=\frac{1}{2}\left(1-\frac{1}{t-1}\right)$ if ${t-1\choose 2}\leq m<{t\choose 2}$
 for some integer $t$.

 For any $r\geq 2$ and any $m\geq 1$, let $C_{r,m}$ be the
 $r$-uniform hypergraph consisting of the first $m$ sets in ${\mathbb{N} \choose r}$ in the colexicographic order (that is $A<B$ if $\max(A\Delta B)\in B$.)
 For example, for $r=3$, the first 5 triple sets under the
 colexicographic order are given below:
 $$\{1,2,3\}<\{1,2,4\}<\{1,3,4\}<\{2,3,4\}<\{1,2,5\}<\cdots$$
 so $C_{3,5}$ is the hypergraph on $5$ vertices with the following edge set
 $$E(C_{3,5})=\{\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},\{1,2,5\}\}.$$

 One can easy to check that if $m={t\choose r}$ for some integer $t$, then
 $C_{r,m}$ is just the complete $r$-uniform hypergraph $K^r_t$ (or $[t]^{(r)}$
 under Talbot's notation \cite{T2002}.)

 In 1989, Frankl and F\"{u}redi made the following conjecture:
 \begin{conj}[{\cite{FF1989}}] \label{conjecture}
   For all $r\geq 3$ and any $m\geq 1$, we have $\lambda_r(m)\leq \lambda(C_{r,m}).$
 \end{conj}

 Talbot \cite{T2002} pointed out that $\lambda(C_{r,m})$ remains  a constant
 ($\lambda(C_{r,m})\equiv{t-1\choose r}/(t-1)^r$)
 for  $m\in [{t-1\choose r}, {t\choose r}-{t-2\choose r-2}]$, and jumps for every
 $m\in [{t\choose r}-{t-2\choose r-2}, {t\choose r}]$.
   Tyomkyn called $m={t\choose r}$
   the principal case. Here we refer the interval $[{t-1\choose r}, {t\choose r}-{t-2\choose r-2}]$ as the {\em principal domain}, and refer $m={t\choose r}-{t-2\choose r-2}$, the {\em critical case}. Most partial results on Frankl-F\"{u}redi's conjecture occur in the principal domain.

   For $r=3$, Talbot \cite{T2002} proved that Conjecture \ref{conjecture} holds whenever ${{t-1}\choose3}\leq m\leq {t\choose 3}-{{t-2}\choose 1}-(t-1)={t\choose3}-(2t-3)$ for some $t\in \mathbb{N}$.
   Tang, Peng, Zhang and Zhao \cite{TPZZ2016}
   extended the interval to ${{t-1}\choose3}\leq m\leq {t\choose 3}-{{t-2}\choose 1}-\frac 1 2(t-1)$.
   Recently, Tyomkyn \cite{T2017} further extended the interval to
   $[{{t-1}\choose 3}, {t\choose 3}- {t-2\choose 1}-\delta_3t^{3/4}]$ for some constant $\delta_3>0$.
   These results can be rephrased in term of the gap (i.e.  the number of missed values)
   in the principal domain: the gap drops from $t-1$, to $\frac{1}{2}(t-1)$, and further to $O(t^{3/4})$. Recently,  Lei, Lu, and Peng \cite{LLP} further reduced the gap to $O(t^{2/3})$.

   For $r\geq 4$, there are fewer results than at $r=3$. In 2017, Tyomkyn \cite{T2017} proved
   the following theorem.
 \begin{thm}[\cite{T2017}]\label{3/4}
 For $r\geq 4$,  there exists a constant $\delta_r>0$ such that
    for any $m$ satisfying ${{t-1}\choose r}\leq m\leq {t\choose r}-{t-2\choose r-2}-\delta_rt^{r-2}$ we have
    $$\lambda_r(m)=\frac{{t-1\choose r}}{(t-1)^r}.$$
 \end{thm}
Here is our main theorem.
  \begin{thm}\label{mainthm}
 For $r\geq 4$,  there exists a constant $\delta_r>0$ such that
    for any $m$ satisfying ${{t-1}\choose r}\leq m\leq {t\choose r}-{t-2\choose r-2}-\delta_rt^{r-\frac{7}{3}}$ we have
    $$\lambda_r(m)=\frac{{t-1\choose r}}{(t-1)^r}.$$
 \end{thm}

Tyomkyn \cite{T2017} proved that the gap can be reduced to $O(t^{r-9/4})$ under an assumption that the hypergraphs have support on $t$ vertices.
We actually proved that the maximum hypergraphs have support on $t$ vertices
for sufficiently large $t$ (see Lemma \ref{keylemma}.)
Moreover, our gap $O(t^{r-7/3})$ improves $O(t^{r-9/4})$ on the exponent slightly.

Another related result is a smooth upper bound on $\lambda_r(m)$. The following result,
which was conjectured (and partially solved for $r=3,4,5$ and any $m$; and for the case $m\geq {4(r-1)(r-2)\choose r}$) by Nikiforov \cite{Nikiforov2018},
was completely proved by the second author.

\begin{thm}[{\cite{maxPspec}}] \label{smooth}
 For all $r\geq 2$ and all $m\geq 1$, if we write $m = {s\choose r}$
   for some real number $s\geq r-1$, then have $$\lambda_r(m)\leq m s^{-r}.$$
  The equality holds if and only if $s$ is an integer and the hypergraph achieving $\lambda_r(m)$ must be the complete  $r$-uniform hypergraph $K^r_s$  (possibly with some isolated vertices added.)
 \end{thm}

The paper will be organized as follows:  the notation and previous lemmas will be given in Section 2. In Section 3, we prove a key lemma that the
maximum hypergraphs have support $t$ for $t$ sufficiently large. Finally, the proof of Theorem \ref{mainthm} will be given in Section 4.

\section{Notation and Preliminaries}
Let $\mathbb{N}$ be the set of all positive integers and $[t]$  the
set of first $t$ positive integers. For any integer $r\geq 2$ and a set $V$, we use $V^{(r)}$ (or ${V\choose r}$) to denote all $r$-subsets of $V$.
An $r$-unform hypergraph (or $r$-graph, for short) consists of a vertex set $V$ and an edge set
$E\subseteq V^{(r)}$.
Given an $r$-graph $G=(V,E)$ and a set $S\subseteq V$ with $|S|<r$, the $(r-|S|)$-uniform {\it link hypergraph} of $S$ is defined as $G_S=(V,E_S)$ with $E_S:=\{f\in V^{(r-|S|)}\colon f\cup S\in E\}$. We will denote the complement graph of $G_S$ by $G^c_S=(V,E^c_S)$ with $E^c_S:=\{f\in V^{(r-|S|)}\colon f\cup S\in V^{(r)}\less E\}$. Define $G_{i\less j}=(V, E_{i\less j})$, where $E_{i\less j}:=\{f\in E_i\less E_j\colon j\notin f\}$ and $G^c_{i\less j}=(V, E^c_{i\less j})$, where $E^c_{i\less j}:=\{f\colon f\cup\{i\}\in E^c ~\text{but}~ f\cup\{j\}\in E\}$, i.e., $E^c_{i\less j}=E^c_i\cap E_j$. Let $G-i$ be the $r$-graph obtained from $G$ by deleting vertex $i$ and those edges containing $i$. A hypergraph $G=(V,E)$ is said to {\it cover} a vertex pair $\{i,j\}$ if there exists an edge $e\in E$ with $\{i,j\}\subseteq e$. The $r$-graph $G$ is said to {\it cover pairs} if it covers every pair $\{i,j\}\subseteq V^{(2)}$.

From now on we assume that $G$ is an $r$-graph on the vertex set $[n]$.
Given a vector $\vec x=(x_1,\ldots,x_n)\in \mathcal{R}^n$,
we write $x_f=x_{i_1}x_{i_2}\cdots x_{i_r}$ if $f=\{i_1,i_2, \ldots, i_r\}$.
The {\it weight polynomial} of $G$ is given by
$$w(G,\vec x)=\sum_{e\in E(G)}x_e.$$
We call $\vec x=(x_1,\ldots,x_n)\in \mathcal{R}^n$  a {\it legal weighting} for $G$ if $x_i\geq0$ for any $i\in[n]$ and $\sum_{i=1}^n x_i=1$.
The set of all legal weightings forms a standard simplex in $\mathbb{R}^n$.
We call a legal weighting ${\vec x}_0$ {\it optimal} if
$w(G,\vec x)$ reaches the maximum at $\vec x={\vec x}_0$ in this simplex. The maximum value
of $w(G,\vec x)$, denoted by $\lambda(G)$,  is called the {\em Lagrangian} of $G$.

\begin{lem}[\cite{FF1989,T2017}]\label{Hmr}
Suppose that $G\subseteq [n]^{(r)}$ and $\overset{\rightarrow}{x}=(x_1,\cdots,x_n)$ is a legal weighting. For all $1\leq i<j\leq n$, we have
\begin{description}
  \item[(i)]  Suppose that $G$ does not cover the pair $\{i,j\}$. Then $\lambda(G)\leq\max\{\lambda(G-i),\lambda(G-j)\}$. In particular, $\lambda(G)\leq\lambda([n-1]^{(r)})$.
  \item[(ii)] Suppose that $m,t\in \mathbb{N}$ satisfy ${{t-1}\choose r}\leq m\leq {t\choose r}-{{t-2}\choose{r-2}}$. Then
  $$\lambda(C_{r,m})=\lambda([t-1]^{(r)})=\frac{1}{(t-1)^r}{{t-1}\choose r}.$$
  \item[(iii)] $w(G_i,\overset{\rightarrow}{x})\leq (1-x_i)^{r-1} \lambda(G_i)$ for any $i\in [n]$.
\end{description}
\end{lem}

Let $E\subset \mathbb{N}^{(r)}$, $e\in E$ and $i,j\in \mathbb{N}$ with $i<j$. We define
\begin{equation*}
L_{ij}(e)=\left\{\begin{array}{cc}
                             (e\less\{j\})\cup \{i\}, & \text{if}~i\notin e ~\text{and}~ j\in e;\\
                             e, & \text{otherwise},
                           \end{array}\right.
\end{equation*}
and $$\mathcal{C}_{ij}(E)=\{L_{ij}(e)\colon e\in E\}\cup \{e\colon e,L_{ij}(e)\in E\}.$$
We say that $E$ is {\it left-compressed} if $\mathcal{C}_{ij}(E)=E$ for every $1\leq i<j$.

From now on, we always assume ${{t-1}\choose r}\leq m< {t\choose r}$ for some integer $t$. Let $G$ be a graph with $e(G)=m$ which satisfies $\lambda(G)=\lambda_r(m)$ and let $\overset{\rightarrow}{x}$ be an (optimal) legal weighting attaining the Lagrangian of $G$. Without loss of generality, we can assume $x_i\geq x_j$ for all $i<j$ and $\overset{\rightarrow}{x}$ has the minimum possible number of non-zero entries, and let $T$ be this number.

Suppose that $G$ achieves a strictly larger Lagrangian than $C_{r,m}$. Then we have

$$\lambda(G)>\frac{1}{(t-1)^r}{{t-1}\choose r},$$

 which in turn implies $T\geq t$, otherwise $\lambda(G)\leq \lambda([t-1]^{(r)})$.

 \begin{lem}[\cite{FF1989,FR1984,T2002}]\label{four}
 Let $G$, $T$, and $\overset{\rightarrow}{x}$ be as defined above. Then
\begin{description}
\item[(i)] $G$ can be assumed to be left-compressed and to cover pairs.
\item[(ii)] For all $1\leq i\leq T$ we have  $$w(G_i,\overset{\rightarrow}{x})=r\lambda(G_i).$$
\item[(iii)] For all $1\leq i<j\leq T$ we have $$(x_i-x_j)w(G_{i,j},\overset{\rightarrow}{x})=w(G_{i\less j},\overset{\rightarrow}{x}).$$
\end{description}
\end{lem}

\begin{lem}[\cite{T2017}] \label{x1} Let $G$, $T$, and $\overset{\rightarrow}{x}$ be as defined above. Then for sufficiently large $t$,
\begin{description}
\item[(i)] $T=t+C$ for some constant $C=C(r)$.

\item[(ii)] $x_1<\frac{1}{t-\alpha}$ for some constant $\alpha=\alpha(r)>0$.
\item[(iii)] If $T=t$, then $x_1<\frac{1}{t-r+1}$.
\end{description}
\end{lem}

Here is our key lemma.

\begin{lem}\label{keylemma}
  Let $G$, $T$, and $\overset{\rightarrow}{x}$ be as defined above. There is a constant $t_0:=t_0(r)$ such that
  if $t\geq t_0$ then $T=t$.
\end{lem}

\section{Proof of Lemma \ref{keylemma}}
We need several lemmas before we prove Lemma \ref{keylemma}.

Suppose that $G$ does not cover the pair $\{i,j\}$. Let $G_{/{ij}}$ be an $r$-uniform hypergraph
obtained from $G$ by gluing the vertices $i$ and $j$ as follows:
\begin{enumerate}
\item Let $v$ be a new fat vertex (by gluing $i$ and $j$.) Then
$G_{/ij}$ has the vertex set $(V(G)\setminus \{i,j\}) \cup \{v\}$.
\item The edge set of $G_{/ij}$ consists of all edges in $G$ not containing $i$ or $j$, plus
the edges of form $\{f\cup \{v\}\colon  f\in E_i \cup E_j\}$.
\end{enumerate}

The following lemma is similar to Lemma \ref{Hmr} (i), but has the advantage of being deterministic.
\begin{lem}\label{G/ij}
 Suppose that $G$ does not cover the pair $\{i,j\}$. Then $\lambda(G)\leq \lambda(G_{/ij})$.
\end{lem}

\begin{proof}
  Let $\vec x$ be an optimal legal weighting of $G$. Define a legal weighting $\vec y$ of $G_{/ij}$ by
$y_v=x_i+x_j$ and $y_k=x_k$ if $k\neq v$.
Then we have
  \begin{align*}
    w(G_{/ij},\vec y)- w(G, \vec x) &= \sum_{f\in E_i\cup E_j} y_fy_v - \sum_{f\in E_i}x_fx_i
- \sum_{f\in E_j}x_fx_j \\
&= \sum_{f\in E_i\cup E_j} x_f (x_i+x_j) - \sum_{f\in E_i}x_fx_i
- \sum_{f\in E_j}x_fx_j \\
&=  \sum_{f\in E_i\setminus E_j} x_fx_j + \sum_{f\in E_j\setminus E_i} x_fx_i\\
&\geq 0.
  \end{align*}
Then we have
$$\lambda(G)=w(G, \vec x)\leq  w(G_{/ij},\vec y)\leq \lambda(G_{/ij}).$$
\end{proof}

\begin{lem} \label{xk}
  For any $k\in[T-1]$, we have
  \begin{equation}\label{eq:xk}
   x_{T-k}>\frac{k-C_0}{k+1}x_1,
  \end{equation}
  where $C_0=C+\alpha-1$.
\end{lem}
\begin{proof}
Observe that
\begin{align*}
1&=x_1+\cdots+x_{T-k-1}+x_{T-k}+\cdots+x_T\\
&< (T-k-1)x_1+(k+1)x_{T-k}\\
&< \frac{T-k-1}{t-\alpha}+(k+1)x_{T-k}.~~~~~~(by~Lemma~ \ref{x1}~ (ii))
\end{align*}
Solving $x_{T-k}$ and applying Lemma \ref{x1} (i) and (ii), we have
$$x_{T-k}>\frac{t-\alpha-T+k+1}{k+1}\cdot\frac{1}{t-\alpha}=
\frac{k-C_0}{k+1}\cdot\frac{1}{t-\alpha}>\frac{k-C_0}{k+1}x_1.$$
\end{proof}

\begin{lem} \label{beta}
There exists a constant $\beta$ such that for any subset $S\subseteq [T]^{(r-2)}$, we have
  \begin{equation}
    \label{eq:sumxk}
    \sum_{f\in S}(x_1^{r-2}-x_f)<\beta x_1.
  \end{equation}
\end{lem}

\begin{proof}
Let $\beta$ be a constant $>\frac{C+\alpha}{(r-3)!}$.
 We will prove it by contradiction.
Suppose that there is $S\subseteq [T]^{(r-2)}$
such that $$\sum_{f\in S}(x_1^{r-2}-x_f)\geq\beta x_1.$$
We have
  \begin{equation}
    \label{eq:S}
  \sum_{f\in S}x_f \leq |S|x_1^{r-2}-\beta x_1.
  \end{equation}
Thus,
\begin{align*}
1&=(x_1+x_2+\cdots+x_T)^{r-2}\\
&\leq(r-2)!\sum_{f\in S}x_f+(T^{r-2}-(r-2)!|S|)x_1^{r-2}\\
&\leq (r-2)!(|S|x_1^{r-2}-\beta x_1) + (T^{r-2}-(r-2)!|S|)x_1^{r-2}~~~~~~~(by~\eqref{eq:S})\\
&= (T x_1)^{r-2}- (r-2)! \cdot \beta x_1.
\end{align*}
On the other hand, note
$\frac{1}{t-\alpha}>x_1\geq \frac{1}{T}=\frac{1}{t+C}$. We have
\begin{align*}
  (T x_1)^{r-2}- (r-2)!\beta x_1
&<\left(1+\frac{C+\alpha}{t-\alpha}\right)^{r-2} - (r-2)!\cdot \beta \frac{1}{t+C}\\
&= 1+ (r-2)\frac{C+\alpha}{t-\alpha} +O(t^{-2}) - (r-2)!\cdot \beta \frac{1}{t-\alpha} +O(t^{-2})\\
&= 1- \frac{(r-2)!}{t-\alpha}\left(\beta- \frac{C+\alpha}{(r-3)!}\right)  +O(t^{-2})\\
&<1.
\end{align*}
Contradiction.
\end{proof}

Let $s=\max\{i\colon \{T-i-(r-2),\ldots,T-i-1,T-i,T\}\in E^c\}$ and $S=\{T-s,T-s+1,\ldots,T-1,T\}$.
We have the following lemma.
\begin{lem} \label{gamma} For $s$ and $S$ defined above, we have
  \begin{enumerate}
  \item Any non-edge $f\in E^c$ must intersect $S$ in at least two elements.
  \item  We have
    \begin{equation}
      \label{eq:XT-1}
    x_{T-1}\geq\gamma x_1,
    \end{equation}
    where $\gamma:=\prod_{k=0}^{r-2}\left(1-\frac{C_0+1}{s+k+1}\right)$.
  \end{enumerate}
\end{lem}
\begin{proof}
By the choice of $s$, we know $\{T-s-(r-2),\ldots,T-s-1,T-s,T\}\in E^c$
but $\{T-s-(r-1),\ldots,T-s-2,T-s-1,T\}\in E$. We now prove item 1 by contradiction.
Suppose not, say there is $f=\{i_1,i_2,\ldots, i_r\}\in E^c$ such that
$i_1<i_2<\cdots<i_{r-1}<T-s$. Since $G$ is left-compressed, $\{T-s-(r-1),\ldots,T-s-2,T-s-1,T\}\in E^c$. Contradiction!

Since  $G$ covers the pair $\{T-1,T\}$, there is an $(r-2)$-tuple
$\{i_1,i_2,\ldots,i_{r-2}\}$ $\in E_{T-1,T}$. We have
 \begin{align}\label{eq:x1xt-1xt}
x_{i_1}\cdots x_{i_{r-2}}x_{T-1}x_T-x_{T-s-(r-2)}x_{T-s-(r-3)}\cdots x_{T-s}x_T\geq 0.
\end{align}
Otherwise by replacing the edge $\{i_1,i_2,\ldots,i_{r-2},T-1,T\}$ with the non-edge $\{T-s-(r-2),T-s-(r-3),\ldots,T-s, T\}$,
we get another $r$-graph with the same number of edges whose Lagrangian is strictly greater than the Lagrangian of $G$. Contradiction!

Combining Inequalities \eqref{eq:x1xt-1xt} and \eqref{eq:xk}, we get
$$x_{T-1}\geq\frac{x_{T-s-(r-2)}x_{T-s-(r-3)}\cdots x_{T-s}}{x_1^{r-2}}> \gamma x_{1}.$$
\end{proof}

We have the following estimation of $\gamma$:
\begin{equation}
  \label{eq:gamma}
\gamma=\prod_{k=0}^{r-2}\left(1-\frac{C_0+1}{s+k+1}\right)=1-\frac{(C_0+1)(r-1)}{s}+ O\left(\frac{1}{s^2}\right).
\end{equation}

\begin{lem}\label{T>t}
  If $T>t$, then $|E^c_{T\backslash (T-1)}| -|E_{T-1,T}|\geq \frac{T-r}{T(r-1)}{T-2 \choose r-2}$.
\end{lem}
\begin{proof} Since $G$ is left-compressed, then
$$|E_T|\leq \frac{rm}{T}\leq \frac{r}{T}{t\choose r}=\frac{t}{T}{t-1\choose r-1}
\leq \frac{t}{T}{T-2\choose r-1}.$$
In the last step, we apply the assumption $t\leq T-1$.
Thus, we have
\begin{align*}
  |E^c_{T\backslash (T-1)}| -|E_{T-1,T}| &= {T-2 \choose r-1}  - (|E_{T\backslash (T-1)}| +|E_{T-1,T}|)\\
&={T-2\choose r-1} - |E_T|\\
&\geq {T-2\choose r-1} - \frac{t}{T}{T-2\choose r-1}\\
&=\frac{T-t}{T} {T-2\choose r-1}\\
&=\frac{(T-t)(T-r)}{(r-1)T}{T-2\choose r-2}\\
&\geq \frac{T-r}{T(r-1)}{T-2 \choose r-2}.
\end{align*}
\end{proof}

\begin{proof}[Proof of Lemma \ref{keylemma}:]

Let $G$, $\vec x$, $m$, $t$, and $T$ be as defined before. Let $\eta:=\left\lceil 2\sqrt {\frac{(C_0+1)(r-1)\beta}{(r-3)!}}t^{r-5/2}\right\rceil$.

We will prove Lemma \ref{keylemma} by contradiction. Assume $T>t$. By Lemma \ref{T>t}, we have
\begin{equation}
  \label{eq:F}
|E^c_{T\backslash (T-1)}| -|E_{T-1,T}|\geq \frac{T-r}{T(r-1)}{T-2 \choose r-2}>\eta.
\end{equation}
This is possible for $t$ sufficiently large since $\frac{T-r}{T(r-1)}{T-2 \choose r-2}=\Theta(t^{r-2})$
and $\eta=\Theta(t^{r-2.5})$.

Let $b=|E_{T-1,T}|$. Pick any $F\subseteq \{f\cup \{T\}\colon f\in E^c_{T\backslash (T-1)}\}$  of size $b+\eta$. This is possible
because of Inequality \eqref{eq:F}.

Let $G'$ be an $r$-graph obtained from $G$ by deleting all edges in $\{f\cup \{T-1,T\}\colon f\in E_{T-1,T}\}$
and adding all $r$-tuples in $F$ as edges.
The main proof is to show the following inequality:
\begin{equation}
  \label{eq:main}
  w(G,\vec x)\leq w(G',\vec x).
\end{equation}

Now we will prove Inequality \eqref{eq:main}. We divide it into two cases.

\noindent
{\bf Case 1:} $\beta s<\eta x_1^{r-3}$. By Lemma \ref{gamma} item 1, every non-edge intersects $S$ in at least two elements. This implies $F$ can be partitioned into $s-1$ parts:
$$F=\cup_{i=2}^s \{f\cup \{T-i, T\}\colon f\in F_i\},$$
where $F_i:=\{f\in [T-i-1]^{(r-2)}\colon f \cup \{T-i,T\} \in F\}$.
We have
\begin{align*}
  w(G',\vec x) &= w(G, \vec x) - \sum_{e\in E_{T-1,T}} x_ex_{T-1}x_T+ \sum_{f\in F} x_f\\
               &\geq w(G, \vec x)+\eta x_1^{r-2}x_{T-1}x_T -\sum_{f\in F}(x_1^{r-2}x_{T-1}x_T-x_f)\\
               &\geq w(G, \vec x)+\eta x_1^{r-2}x_{T-1}x_T - \sum_{i=2}^s
                 \sum_{f'\in F_i}(x_1^{r-2}-  x_{f'})  x_{T-1}x_T\\
                &\geq w(G, \vec x)+\eta x_1^{r-2}x_{T-1}x_T - x_{T-1}x_T\sum_{i=2}^s \beta x_1   ~~~~~~~~~~~~~~~~~~~(\mbox{by}~ \eqref{eq:sumxk})\\
                &>w(G, \vec x)+ \left(\eta x_1^{r-3}-s\beta\right)x_1x_{T-1}x_T\\
                &>w(G, \vec x).
\end{align*}

\noindent
{\bf Case 2:} $\beta s\geq \eta x_1^{r-3}$.  Since $x_1\geq \frac{1}{T}$, we have
\begin{equation}
  \label{eq:s}
s\geq \frac{\eta}{\beta} x_1^{r-3}\geq \frac{\eta}{\beta} \frac{1}{T^{r-3}}.
\end{equation}

We first prove an inequality:
\begin{equation}
  \label{eq:gammab}
   \gamma^{r-2}\eta > (1-\gamma^{r-2})b.
\end{equation}
\begin{align*}
&\hspace*{-5mm}  \gamma^{r-2}\eta - (1-\gamma^{r-2})b\\
  &\geq  \gamma^{r-2}\eta - (1-\gamma^{r-2}){T-2\choose r-2}\\
  &>\gamma^{r-2} \left( \eta- (\gamma^{-(r-2)} -1) \frac{T^{r-2}}{(r-2)!} \right)\\
  &>\gamma^{r-2} \left( \eta- \left(\frac{(C_0+1)(r-1)(r-2)}{s} + O\left(\frac{1}{s^2}\right) \right)
    \frac{T^{r-2}}{(r-2)!} \right)
    ~~~~~(\mbox{by}~ \eqref{eq:gamma})\\
  &=\frac{\gamma^{r-2}}{s}  \left( s\eta- \left((C_0+1)(r-1)(r-2) + O\left(\frac{1}{s}\right) \right)
    \frac{T^{r-2}}{(r-2)!} \right)~~(\mbox{using}~\eqref{eq:s})\\
  &\geq\frac{\gamma^{r-2}}{s}  \left(\frac{1}{\beta T^{r-3}}\eta^2- \left((C_0+1)(r-1)(r-2) + O\left(\frac{1}{s}\right) \right)
    \frac{T^{r-2}}{(r-2)!} \right)\\
  &>0.
\end{align*}
In the last step, we apply the definition of $\eta $ and get
$$\eta^2\geq 4 \frac{(C_0+1)(r-1)(r-2)\beta}{(r-2)!}t^{2r-5}>
2 \frac{(C_0+1)(r-1)(r-2)\beta}{(r-2)!} T^{2r-5}.$$
Now we are ready to estimate $ w(G',\vec x)$:
\begin{align*}
  w(G',\vec x)  &= w(G, \vec x) - \sum_{e\in E_{T-1,T}} x_ex_{T-1}x_T+ \sum_{f\in F} x_f\\
                &>w(G, \vec x)-b x_1^{r-2}x_{T-1}x_T + (b+\eta) x_{T-1}^{r-1}x_T\\
                &=w(G, \vec x)+x_{T-1}x_T\left((b+\eta) x_{T-1}^{r-2}-b x_1^{r-2}\right)\\
                &\geq w(G, \vec x)+ x_{T-1}x_T\left((b+\eta) \gamma^{r-2} x_1^{r-2}- b x_1^{r-2}\right)~~~~~~(by~\eqref{eq:XT-1})\\
                &=w(G, \vec x)+ x_1^{r-2}x_{T-1}x_T\left(\gamma^{r-2}\eta -(1-\gamma^{r-2})b\right)~~~~~~(by~\eqref{eq:gammab})\\
               &> w(G, \vec x).
\end{align*}
Therefore, Inequality \eqref{eq:main} holds in any circumstances.

Note that  $G'$ does not cover the pair $\{T-1, T\}$.
Applying Lemma \ref{G/ij}, we have
$$\lambda(G)=w(G, \vec x)<  w(G',\vec x)\leq \lambda(G')\leq \lambda(G'_{/(T-1)T}).$$
By the construction of $G'$, the added edges  are from $F\subseteq \{f\cup \{T\}\colon f\in E^c_{T\backslash (T-1)}\}$.
These edges have the form of $f\cup \{T\}$, where $f\cup \{T-1\}$ is also an edge in $G$.
After gluing $T-1$ and $T$ together, both edges $f\cup \{T\}$ and $f\cup \{T-1\}$ are glued into one edge $f\cup \{v\}$. We have
$$|E(G'_{/(T-1)T})|\leq |E(G')|-|F|\leq |E(G)|.$$
Contradiction! This completes the proof of Lemma~\ref{keylemma}.

\end{proof}

\section{Proof of Theorem \ref{mainthm}}
Assume $t\geq t_0$.  Let $G=(V,E)$ be an $r$-graph with $m$ edges satisfying ${{t-1}\choose r}\leq m\leq {t\choose r}-{t-2\choose r-2}$ and $\lambda(G)=\lambda_r(m)>\lambda([t-1]^{(r)})$.
Let $\vec x=(x_1,\ldots, x_n)$ be an optimal legal weighting for $G$ that uses exactly $T$ nonzero weights (i.e., $x_1\geq \cdots \geq x_T> x_{T+1}=\cdots =x_n=0$).  By Lemma \ref{keylemma}, we may assume $T=t$. In addition, we may assume $G$ is left-compressed
by Lemma \ref{four}(i).

Since $T=t$, by Lemma \ref{x1}(iii), we have
\begin{equation}
  \label{t:x1}
x_1<\frac{1}{t-r+1},
\end{equation}

and we may improve Lemma \ref{xk} as follows.
\begin{lem} \label{xkt}
  For any $k\in[t-1]$, we have
  \begin{equation}\label{eq:xkt}
   x_{t-k}>\frac{k-r+2}{k+1}x_1.
  \end{equation}
\end{lem}
Let $T=t$, $C=0$, $\alpha=r-1$, $C_0=r-2$, and $\beta=\frac{r}{(r-3)!}$ in the proof of Lemma \ref{beta}. Then we can get the following Lemma improving Lemma \ref{beta}.
\begin{lem} \label{betat}
For any subset $S\subseteq [t]^{(r-2)}$, we have
  \begin{equation}
   \label{t:beta}
    \sum_{f\in S} (x_1^{r-2}-x_f)<\frac{r}{(r-3)!} x_1.
  \end{equation}
\end{lem}

With $T=t$, we have $s=\max\{i\colon \{t-i-(r-2),\ldots,t-i-1,t-i,t\}\in E^c\}$ and $S=\{t-s,t-s+1,\ldots,t-1,t\}$. Lemma \ref{gamma} can be improved to:
\begin{lem} \label{gammat} For $s$ and $S$ defined above, we have
  \begin{enumerate}
  \item Any non-edge $f\in E^c$ must intersect $S$ in at least two elements.
  \item  We have
    \begin{equation}
      \label{eq:xt-1}
    x_{t-1}>\gamma x_1,
    \end{equation}
    where $\gamma:=\prod_{k=0}^{r-2}\left(1- \frac{r-1}{s+k+1}\right)$.
  \end{enumerate}
\end{lem}

\begin{lem} \label{minimum}
Let $\gamma$ be defined in Lemma \ref{gammat}. Consider two functions:
  \begin{align}
    g_1(s)&:={s+1\choose 2}\frac{r\gamma^{-(r-2)}}{(r-3)!}t^{r-3},\\
    g_2(s)&:={t-2\choose r-2}(\gamma^{-(r-2)}-1).
  \end{align}
  Then
  \begin{equation}
  \label{eq:eta}
  \min\{g_1(s),g_2(s)\}=O(t^{r-\frac7 3}).
\end{equation}
\end{lem}
\begin{proof}
 By Equation \eqref{eq:gamma} (with $C_0=r-2$),   we have
  $$\gamma=1-\frac{(r-1)^2}{s}+O\left(\frac{1}{s^2}\right).$$
  When $s\leq t^{1/3}$, we have
  $$g_1(s)={s+1\choose 2}\frac{r\gamma^{-(r-2)}}{(r-3)!}t^{r-3} =O(t^{r-7/3}).$$
  When $s\geq t^{1/3}$, we have
  \begin{align*}
    g_2(s)&={t-2\choose r-2}(\gamma^{-(r-2)}-1)\\
          &= {t-2\choose r-2}\left(\frac{(r-1)^2(r-2)}{s}+O\left(\frac{1}{s^2}\right)\right)\\
            &=O(t^{r-7/3}).
  \end{align*}
  Thus, Equality \eqref{eq:eta} holds.
\end{proof}

\begin{proof}[Proof of Theorem \ref{mainthm}:]
  We assume $t\geq t_0$ so that $T=t$ holds. Let $\eta:=\lceil \min\{f(s),g(s)\}\rceil =O(t^{r-7/3})$.

\noindent {\bf Claim\refstepcounter{counter}\label{mainclaim}  \arabic{counter}.} For  $t\geq t_0$, if $ {t-1\choose r}\leq m\leq {t\choose r}-{t-2\choose r-2}-\eta$, then $\lambda_r(m)=\frac{{{t-1}\choose r}}{(t-1)^r}$.

We will prove this claim by contradiction. Assume there is a graph $G$ with $m$ edges, where $ {t-1\choose r}\leq m\leq {t\choose r}-{t-2\choose r-2}-\eta$, and $\lambda(G)=\lambda_r(m)>\frac{{{t-1}\choose r}}{(t-1)^r}$.
Let $B$ be any family of $|E_{t-1,t}|$ many non-edges of $G$ which does not contain both $t$ and $t-1$.
This is possible since $G$ has at least ${{t-2}\choose{r-2}}+\eta$ non-edges.
Let $G'$ be an $r$-graph obtained from $G$ by deleting all edges in $E_{t-1,t}$
and adding all $r$-tuples in $B$ as edges. Then $G'$ has exactly $m$ edges.

By Lemma \ref{gammat} item 1, any non-edge in $B$ must intersect $S$ in at least two elements.
For any $\{i,j\}\subseteq S^{(2)}$ with $i<j$, define
$$B_{ij}:=\{\{i_1,\ldots,i_{r-2}\}
\colon \{i_1,\ldots,i_{r-2},i,j\}\in B
~\text{and}~ i_1<\cdots<i_{r-2}<i<j\}.$$
Then we have $$B= \bigcup\limits_{\{i,j\}\subseteq S^{(2)}}\{f\cup \{i,j\}\colon f\in B_{ij}\}.$$
We allow some $B_{ij}$ to be emptysets. (For example, $B_{t-1,t}=\emptyset$.)

Now, we consider the difference between $w(G',\vec x)$ and $w(G, \vec x)$.
On the one hand,
\begin{align*}
  w(G',\vec x) &= w(G, \vec x) - \sum_{e\in E_{t-1,t}} x_ex_{t-1}x_t+ \sum_{f\in B} x_f\\
               &\geq w(G, \vec x)- \sum_{f\in B}(x_1^{r-2}x_{t-1}x_t-x_f)\\
               &\geq w(G, \vec x)-\sum_{\{i,j\}\in S^{(2)}}\sum_{f'\in B_{ij}}(x_1^{r-2}-x_{f'})x_{t-1}x_t\\
                &>w(G, \vec x)- {s+1\choose 2}\frac{r}{(r-3)!}x_1x_{t-1}x_t ~~~~~~~~~~~~~~~~~~~~~~~~(by~ \eqref{t:beta})\\
                &>w(G, \vec x)- {s+1\choose 2}\frac{r}{(r-3)!}t^{r-3}x_{1}^{r-2}x_{t-1}x_t.  ~~~~~~~~(\mbox{since } x_1\geq \frac{1}{t})\\
               &>w(G, \vec x)- {s+1\choose 2}\frac{r\gamma^{-(r-2)}}{(r-3)!}t^{r-3}x_{t-1}^{r-1}x_t.  ~~~~~~~~~~~~~~~~~~~(by~ \eqref{eq:xt-1})\\
  &=w(G, \vec x)- g_1(s) x_{t-1}^{r-1}x_t.
\end{align*}

On the other hand,
 \begin{align*}
   w(G',\vec x) &= w(G, \vec x) - \sum_{e\in E_{t-1,t}} x_ex_{t-1}x_t+ \sum_{f\in B} x_f\\
    &> w(G, \vec x)- \sum_{f\in B}(x_1^{r-2}x_{t-1}x_t-x_{t-1}^{r-2}x_{t-1}x_t)\\
               &> w(G, \vec x)-{t-2\choose r-2}(x_1^{r-2}-x_{t-1}^{r-2})x_{t-1}x_t\\
               &=w(G, \vec x)- {t-2\choose r-2}(\gamma^{-(r-2)}-1)x_{t-1}^{r-1}x_t ~~~~~~~~~~~~~~~~~~~~~~~~~~~~(by~ \eqref{eq:xt-1})\\
    &=w(G, \vec x)- g_2(s) x_{t-1}^{r-1}x_t.
\end{align*}
Thus, we have
\begin{equation}
  \label{eq:diff}
   w(G',\vec x) >  w(G, \vec x) -\min\{g_1(s), g_2(s)\} x_{t-1}^{r-1}x_t \geq w(G, \vec x) - \eta x_{t-1}^{r-1}x_t.
\end{equation}

Note that $G'$ does not cover $\{t-1,t\}$ and $G'$ still has $\eta$ non-edges which does not contain
both $t-1$ and $t$. Let $G''$ be an $r$-graph obtained from $G'$ by adding these $\eta$ $r$-tuples
as edges. We have
\begin{align*}
  w(G'',\vec x) &\geq w(G',\vec x) +\eta x_{t-r+1}x_{t-r+2}\cdots x_t\\
                &> w(G, \vec x) - \eta x_{t-1}^{r-1}x_t + \eta x_{t-r+1}x_{t-r+2}\cdots x_t\\
                &>w(G, \vec x).
\end{align*}
Note that $G''$ still does not cover the pair $\{t-1,t\}$. We have
 $$\lambda(G)=w(G,\vec x) <w(G'',\vec x)\leq \lambda(G'')\leq\lambda([t-1]^{(r)}),$$
a contradiction. Claim \ref{mainclaim} is proved.

Finally we can choose a constant $\delta_r$ large enough so that the following two conditions hold:
\begin{itemize}
\item $\delta_r t^{r-7/3}>\eta$ for all $t\geq t_0$,
\item and $\delta_r t^{r-7/3}>{t-2 \choose r-1}$ for $1\leq t\leq t_0.$
\end{itemize}
When $t\geq t_0$, we have
$$\lambda_r(m)\leq \lambda\left(C_{r, {t\choose r}-{t-2\choose r-2}-\delta_rt^{r-7/3}}\right) \leq \lambda\left(C_{r,{t\choose r}-{t-2\choose r-2}-\eta}\right)=\frac{{t-1\choose r}}{(t-1)^r}.$$
When $1\leq t\leq t_0$, we have
$$\lambda_r(m)\leq \lambda\left(C_{r,{t-1\choose r}}\right)= \frac{{t-1\choose r}}{(t-1)^r}.$$
Since $m\geq {t-1\choose r}$, we have
$$\lambda_r(m)\geq \lambda_r\left({t-1\choose r}\right)=\frac{{t-1\choose r}}{(t-1)^r}.$$
Thus, $$\lambda_r(m)=\frac{{t-1\choose r}}{(t-1)^r}.$$
This completes the proof of Theorem~\ref{mainthm}.

  \end{proof}

config: arxiv


TITLE: Rational solutions of $x^4(k^2+4)-4kx^2=z^2$, where $k>0$ is given rational.
QUESTION [0 upvotes]: How to generate some parametric family of rational solutions to $x^4(k^2+4)-4kx^2=z^2$, where $k>0$ is given rational.

REPLY [1 votes]: For $k=4$, Above equation is shown below:
$20x^4=(4x)^2+z^2$  ------(A)
Equation $(A)$ has parametric solution:
$x=2p(5m^2-4m+1)$
$z=16(p^2)(25m^4-45m^3+30m^2-9m+1)$
Where, $p=[1/(5m^2-1)]$
For $m=3$ we have:
$(x,z,k)=[(17/11),(1054/121),(4)]$


TITLE: Do physical entities with one dimension equal to 0 exist?
QUESTION [5 upvotes]: Within mathematics, a lot of entities with one or more size equal to zero exist and are fluently manipulated.
For example, within $\mathbb R^4$, the entity defined as $z=0, t=2$ is a plan, the one defined as $x=ct, y=0, z=0$ is a line, and the one defined as $x=0, y=0, z=0, t=0$ is a point.
Do this mathematical entities have a real counterpart?
For example does there exist any physical entity which one spatial dimension would equal 0? By 0 I mean here a real zero (not $0 \in \mathbb R$, and not $0±10^{-15}$) which might be unreachable to us due to the Heisenberg's uncertainty principle, but nonetheless physical and even provable due to its consequences.
For example 2 such objects with let's say $\delta x = 0$, could occupy the same spatial position $x=x_0$ at the exact same time $t=t_0$.
As a typical example, the electron is said to be a "point like" entity. This expression seems to have been carefully chosen to avoid as a bad word the expression of a "null size" entity.
My naive point of view is that null size "can't exist" because it leads to:

infinite physical densities,
impossibility to disambiguate 2 such null entities (2 particles of null size, with all other physical characteristics identical, can occupy the same position at the same time, and nothing can help to name one out of the 2),
infinite number of such entities in the exact same position.

Is the existence of such entities proven within the field of elementary particles or electromagnetic waves?

REPLY [1 votes]: My naive point of view is that null size "can't exist" because it leads to:
infinite physical densities,

In classical physics; the underlying level of nature is quantum mechanical , as far as our experiments and the theory that describes them. There are no infinite densities.

impossibility to disambiguate 2 such null entities (2 particles of null size, with all other physical characteristics identical, can occupy the same position at the same time, and nothing can help to name one out of the 2),

It would be true classically but  not quantum mechanically,  because there exist quantum numbers that are conserved and separate particles . Elementary particles are point like,  quantum mechanical entities whose interactions can be predicted using the mathematics of Feynman diagrams. For example in Compton scattering

A photon and an electron meet at a mathematical point , an integral over the variables will give the probability density for the outgoing photon and electron. The whole process in real space is bounded by the Heisenberg uncertainty, which is inherent in the quantum mechanical mathematical model. Thus, the model has point particles and overlaps, the measurements have an inherent quantum mechanical uncertainty.

infinite number of such entities in the exact same position.

The very successful quantum mechanical theory, gives point particles which have  3 point vertices only, so no infinities . One has to understand that one has to develop a new intuition for quantum mechanics. One cannot extrapolate classical physics behavior to the elementary particles.
All macroscopic physical entities are constructed by elementary particles, and their dimensions are limited by the Heisenberg uncertainty and the statistical distributions of many bodies, so until one reaches the elementary particle level there are always three dimensions to them. At the elementary particle level the answer is yes, at the theoretical level, bounded by the HUP at measurement level.


TITLE: Terminology for line integral of magnetic field
QUESTION [3 upvotes]: One of the quantities appearing in the integral form of Maxwell's Equations is the line integral of the magnetic field around a closed loop.  (The relevant equation states that this is equal to the current through any surface bounded by that loop, plus the displacement current in the case of changing electric fields, with some constant coefficients possibly thrown in, depending on how you manage your units.  This is usually called Ampère's Law, with Maxwell's correction for the displacement current.)
Is there a name for this quantity?  More generally, is there a name for the line integral of the magnetic field along an arbitrary non-closed curve?  (Then this is not equal to any other named quantity or sum of quantities.)  Ideally, I'd want a name for the integral of the H-field (rather than the B-field, when one distinguishes them) in amperes (or the equivalent in non-SI units), but I am not really picky about those details.
It seems that every other quantity in the integral form of Maxwell's Equations has a name (magnetic flux, electric flux, charge, etc), so I'd hope that this one does too.  Of course, the term for the line integral of the electric field (‘electromotive force’) is somewhat of a historical oddity, so maybe this quantity is too obscure to have a name.  Still, you'd think that somebody would have given it one sometime!
ETA:  A theoretical way to measure this quantity is to place a conductive tube (as conductive as possible, with as small diameter as possible) around the curve in question.  The magnetic flux through the tube will induce a current running around (not along) the inside of the tube, which will in turn serve to cancel (or shield) the source of the magnetizing field, in accordance with Lenz's Law (in the broad sense).  The current so induced, serving to completely cancel the magnetic field near the curve in question, should equal this quantity, if I'm thinking about this correctly.  (Well, I'm having a little trouble getting the sign straight, but one way or the other it should work!)

REPLY [0 votes]: The English Wikipedia article for reluctance uses the term ‘magnetomotive force’.  I like Tobias's answer better, so I'm accepting it, but I'm also recording this one.  If somebody else adds another better answer, then maybe I'll switch!


text:
\begin{document}

\baselineskip 16pt

\title{Finite groups with Frobenius normalizer condition for non-normal
 primary subgroups  \thanks{Research  is supported by
 China Scholarship Council,  NNSF of
China(11771409) and Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences.}
}

\author{Zhang Chi , Wenbin Guo\thanks{Corresponding author}\\
{\small Department of Mathematics, University of Science and
Technology of China,}\\ {\small Hefei 230026, P. R. China}\\
{\small E-mail:
zcqxj32@mail.ustc.edu.cn, wbguo@ustc.edu.cn}}

\date{}
\maketitle

\date{}
\maketitle

\begin{abstract}   A finite group  $P$ is said to be
\emph{primary }  if $|P|=p^{a}$ for some prime $p$.
 We say a primary subgroup  $P$ of a finite group $G$
 satisfies   the
\emph{Frobenius normalizer condition} in $G$
         if  $N_{G}(P)/C_{G}(P)$  is a $p$-group provided $P$ is $p$-group.

In this paper, we determine the structure of a finite group $G$  in which
  every non-subnormal primary
 subgroup
 satisfies    the   Frobenius normalized condition.  In particular, we
prove that if    every non-normal primary
 subgroup of $G$
 satisfies    the   Frobenius condition, then $G/F(G)$ is cyclic and every
maximal non-normal nilpotent subgroup $U$ of $G$ with $F(G)U=G$  is a Carter subgroup
of $G$.

\end{abstract}

\footnotetext{Keywords: finite group, nilpotent group, soluble group,  Carter subgroup,
Frobenius normalizer condition.
 }

\footnotetext{Mathematics Subject Classification (2010): 20D10,
20D15, 20D30}
\let\thefootnote\thefootnoteorig

\section{Introduction}

Throughout this paper, all groups are finite and $G$ always denotes
a finite group. Moreover,  $\mathbb{P}$ is the set of all    primes,
$p\in \pi \subseteq  \Bbb{P}$ and  $\pi' =  \Bbb{P} \setminus \pi$; $p'=\mathbb{P}\setminus \{p\}$. If $n$ is an integer,
 the symbol $\pi (n)$
denotes the
 set of all primes dividing $n$; as usual,  $\pi (G)=\pi (|G|)$, the set of all
 primes dividing the order of $G$. Recall also that $G$ is said to be
\emph{primary }  if $|G|=p^{a}$ for some prime $p$.  A normal subgroup $N$
of $G$ is said to be \emph{hypercentral in $G$} if either $N=1$ or every
chief factor $H/K$ of $G$ below $N$ is central, that is, $C_{G}(H/K)=G$.
The  product of all hypercentral subgroups of $G$  is called the
\emph{hypercentre}  of $G$ and denoted be $Z_{\infty}(G)$.

The nature of the embedding  of primary subgroups in the group has a
significant effect on the structure of this group (see
 the books \cite{prod, GuoII} and the surveys in  \cite{Ore, commun}).
Recall, for example,  that by  the well-known  Frobenius theorem
\cite[IV, Satz 5.8]{hupp},
 $G$ is $p$-nilpotent
 if and only if for every $p$-subgroup $P$ of $G$
the section
 $N_{G}(P)/C_{G}(P)$  is a $p$-group too.

{\bf Definition 1.1.}      We say that a primary subgroup  $P$ of $G$
 satisfies   the
\emph{Frobenius normalizer condition} in $G$
         if  $N_{G}(P)/C_{G}(P)$  is a $p$-group provided $P$ is a $p$-group.

Thus $G$ is nilpotent if and only if every  primary
 subgroup  of $G$
 satisfies   the
Frobenius normalizer condition in $G$.

Before continuing, consider the following example.

{\bf Example 1.2.} (i) Let $p  >  q  > r > t$ be primes, where $q$ divides $p-1$ and
 $t$ divides $r-1$.
Let  $C_{p}\rtimes
C_{q}$  be
  a non-abelian group of order $pq$, $C_{r}\rtimes C_{t}$  be
  a non-abelian group of order $rt$,  and  let $G=(C_{r}\rtimes C_{t}) \times (C_{p}\rtimes
C_{q})$.  Then  every non-normal primary  subgroup of $G$
 satisfies    the
Frobenius normalizer condition in $G$.

(ii) Recall that $G$ is called    \emph{semi-nilpotent} (see
  Section 7  in
 \cite[Ch. 4]{We} if
the normalizer of every non-normal nilpotent subgroup of $G$ is nilpotent.
It is clear that in every semi-nilpotent group $G$, every non-normal primary
 subgroup  satisfies   the Frobenius normalizer condition in $G$. The converse is not true
 in general case (see the group $G$ in Part (i)).

The Example  1.2(i) shows that groups in which every non-normal primary
 subgroup
 satisfies   the
Frobenius normalizer condition may be non-nilpotent.
Nevertheless,  our first  result shows that the groups with such a property
 have the structure very close
to  the structure of nilpotent groups.

{\bf Theorem 1.3. } {\sl If    every non-normal primary
 subgroup of $G$
 satisfies    the   Frobenius normalizer  condition, then  $G/F(G)$ is cyclic and
 all maximal nilpotent
subgroups $U$ of $G$ with $F(G)U=G$ are Carter subgroups of $G$.  }

We prove  Theorem 1.3 being based on the following our general result.

{\bf Theorem 1.4. } {\sl Suppose that   every non-subnormal  primary
 subgroup of $G$
 satisfies    the   Frobenius normalizer condition. Then $G$ is either of the following type:}

(a) {\sl $G$ is nilpotent.}

(b) {\sl $G$ is soluble and $G$ has a Sylow basis $P_{1}, \ldots ,  P_{t}$
 such that:}

(i) {\sl  For some $1\leq r < t$ the subgroups
$P_{1}, \ldots ,  P_{r}$ are normal in $G$,  $P_{i}$ is not   normal in
$G$ for all $i > r$
 and $E= P_{r+1} \cdots  P_{t} $  is nilpotent.}

(ii) {\sl $F(G)$ is a maximal
 nilpotent subgroup of $G$  and
  $F(G)=F_{0}(G)Z_{\infty}(G)$, where
 $F_{0}(G)=P_{1} \cdots   P_{r}$.}

(iii) {\sl $N_{G}(E)$ is a Carter subgroup  of $G$.}

(iv) {\sl $V_{G}= Z_{\infty}(G)$
for every maximal nilpotent subgroup $V$ of $G$ such that  $G=F(G)V$.
}

{\sl Conversely, if $G$ is a group of  type (a) or (b), then
every non-subnormal  primary       subgroup of $G$
 satisfies    the   Frobenius normalizer condition. }

In view of Example  1.2(ii) we get from Theorems  1.4  the following

{\bf Corollary 1.5} (See   Theorem 7.6 in \cite[Ch. 4]{We}).  {\sl If $G$ is
 semi-nilpotent and $F_{0}(G)$ denotes the  product of its normal
 Sylow subgroups, then
$G/F_{0}(G)$ is nilpotent. }

{\bf Corollary 1.6} (See   Theorem 7.8 in \cite[Ch. 4]{We}).  {\sl If $G$ is
semi-nilpotent, then: }

(a) {\sl $F(G)$ is a maximal nilpotent subgroup of $G$.}

(b) {\sl If $U$ is  a maximal nilpotent subgroup of $G$ and $U$ is not normal in $G$, then
 $U_{G}=Z_{\infty}(G)$.}

{\bf Proof.}  If $U$ is  a maximal nilpotent subgroup of $G$ and $U$ is not normal in $G$, then
$N_{G}(U)=U$, so $U$ is a Carter subgroup of $G$. Hence we get both the Statements (a) and (b)
 from Theorem 1.4.

 From Theorems 1.3 we get   the following

{\bf Corollary 1.7} (Chih-Han Sah  \cite{Sah}).  {\sl If $G$ is
semi-nilpotent, then  $G/F(G)$ is cyclic.    }

\section{Proof of Theorem 1.4}

Recall that  $G$ is called a \emph{Schmidt group} if $G$ is not nilpotent
but
 every proper subgroup of $G$ is nilpotent.

 {\bf Lemma 2.1} (See  \cite[Ch. 1, Proposition 1.9]{{GuoII}}).  {\sl If
$G$ is a
 Schmidt group, then $G=P\rtimes Q$, where    $P=G'$
 is a Sylow $p$-subgroup of $G$ and $Q$ is a
 Sylow $q$-subgroup of $G$ for some primes $p\ne q$.    }

$G$ is said to be \emph{$p$-decomposable} if $G=O_{p}(G)\times
O_{p'}(G)$.

 {\bf Lemma  2.2.}  {\sl Suppose that $G$ is soluble and
 let
$P_{1}, \ldots , P_{t}$ be  a Sylow basis of $G$,
 where $P_{i}$ is a
 $p _{i}$-group for all $i$.  If $N_{G}(P_{i})$  is
 a $p _{i}$-decomposable for  all  $i=1, \ldots t$,
 then $G$ is nilpotent.  }

 {\bf Proof. }   Let $R$ be a minimal normal
subgroup of $G$. Then $R$ is a $p _{k}$-group for some $k$ since $G$
is  soluble by hypothesis.
Moreover,  $P_{1}R/R, \ldots , P_{t}R/R$ is  a Sylow basis of $G/R$.
It is clear also that   $N_{G}(P_{i})R=N_{G}(P_{i}R)$, so
 $$N_{G/R}(P_{i}R/R)=N_{G}(P_{i})R/R \simeq N_{G}(P_{i})/N_{G}(P_{i})\cap R$$
is $p_{i}$-decomposable for all $i$. Therefore the hypothesis holds for $G/R$, so $G/R$
is  nilpotent  by induction. Hence $P_{k}/R$ is normal in $G/R$, so
 $G=N_{G}(P_{k})$  is $p _{k}$-decomposable. Let $N$ be a minimal
normal subgroup of $G$ contained in $O_{p_{k}'}(G)$. Then  $G/N$ is
nilpotent, so $G\simeq G/1=G/R\cap N$ is  nilpotent.
  The lemma    is proved.

{\bf Lemma 2.3.} {\sl  Let $H$, $E$ and $N$ be subgroups of $G$, where
 $H$ is a   $p$-subgroup of $E$ and $N$ is a  soluble normal subgroup of  $G$.
 If $N_{G}(H)/C_{G}(H)$ is a  $p$-group   and either $N\leq H$ or $(|N|, |H|)=1$, then
 $N_{G/N}(NH/N)/C_{G/N}(HN/N)$  and $N_{E}(H)/C_{E}(H)$ are $p$-groups.}

{\bf Proof. }  First  assume that $(|N|, |H|)=1$. Then $N_{G}(NH)=NN_{G}(H)$,    so
 $$N_{G/N}(NH/N)/(C_{G}(H)N/N)= (N_{G}(NH)/N)/(C_{G}(H)N/N)=
  (N_{G}(H)N/N)/(C_{G}(H)N/N)$$$$
\simeq N_{G}(H)N/C_{G}(H)N\simeq N_{G}(H)/(N_{G}(H)\cap NC_{G}(H))=
N_{G}(H)/C_{G}(H)(N_{G}(H) \cap N) $$$$\simeq (N_{G}(H)/C_{G}(H))/(C_{G}(H)(N_{G}(H)
 \cap N) /C_{G}(H))$$  is  $p$-group, where
$C_{G}(H)N/N\leq C_{G/N}(NH/N)$. Hence  $N_{G/N}(NH/N)/C_{G/N}(NH/N)$  is a
$p$-group.
  Similarly one can shows that  $N_{G/N}(H/N)/C_{G/N}(H/N)$  is a
$p$-group in the case when $N\leq H$.

Finally,
$$N_{E}(H)/C_{E}(H)=
 (N_{G}(H)\cap E)/(C_{G}(H)\cap E)\simeq (N_{G}(H)\cap
E)C_{G}(H)/C_{G}(H)$$ is a   $p$-group.
The lemma is proved.

 {\bf Lemma 2.4.} {\sl  If $A$ is a subnormal nilpotent $\pi$-subgroup of
$G$, then $A\leq O_{\pi}(G)\cap F(G)$.}

{\bf Proof. }   There is a subgroup chain  $A=A_{0} \leq A_{1} \leq \cdots \leq
A_{n}=G$  such that  $A_{i-1}\trianglelefteq A_{i}$   for all $i=1, \ldots , n$.
We can assume  without loss of generality that $M=A_{n-1}  < G$.  Then by
induction we have that $A\leq O_{\pi}(M)\cap F(M)$. On the other hand, the
subgroups $O_{\pi}(M)$ and $F(M)$ are normal in $G$ since they are
characteristic in $M$, so  $O_{\pi}(M)\leq O_{\pi}(G)$ and $F(M)\leq
F(G)$. The lemma is proved.

{\bf Proposition 2.5.} {\sl  Suppose that for any  Sylow $p$-subgroup
 $P$  of $G$,     every
non-subnormal subgroup  $H$ of $G$ contained in the focal subgroup $G'\cap P$
 satisfies    the   Frobenius  normalizer
condition in $G$.   Then $G$ is $p$-soluble.}

{\bf Proof. } Assume that this proposition is false and let $G$ be
 a counterexample of minimal order. Then $p\in \pi (G)$.

(1) {\sl If $A$ is any $p$-closed Schmidt
subgroup of $G$,  where $p$ divides $|A|$, then
for some minimal normal subgroup $R$ of $G$ we have $R\leq O_{p}(G)$. }

Let $A_{p}$ be the normal Sylow subgroup of $A$. Then $A_{p}=A'\leq G'\cap
G_{p}$ for some Sylow $p$-subgroup $G_{p}$ of $G$ by Lemma 2.1.  On the other hand,
$A_{p}$
 does not satisfy the
 Frobenius normalizer condition in $G$ since $A\leq N_{G}(A_{p}) $   and
$A$ is not $p$-nilpotent. Therefore $A_{p}$ is subnormal in $G$ by
hypothesis, so $A_{p}\leq O_{p}(G)$ by Lemma 2.4.  Hence we have (1).

(2)  {\sl  Every maximal subgroup of $G$ is $p$-soluble.}

First we show that the hypothesis holds for every subgroup $E$ of $G$.
Indeed, let $p\in   \pi(E)$ and let $H$ be any  non-subnormal  $p$-subgroup of $E$ such that
for some Sylow $p$-subgroup $E_{p}$ of $E$ we have $H\leq E'\cap E_{p}$.
Then for some Sylow $p$-subgroup $G_{p}$ of $G$ we have $E_{p}\leq G_{p}$.
On the other hand, $H$ is not subnormal in $G$ by \cite[Ch. A, Lemma 14.1(a)]{DH} since
 it  not subnormal in $E$  and
$H\leq G'\cap G_{p}$ since $E'\leq G'$. Hence  $H$
  satisfies    the   Frobenius normalizer  condition
 in $G$ by hypothesis and hence $H$
  satisfies    the   Frobenius normalizer condition in $E$ by Lemma 2.3.  Therefore
the hypothesis holds for $E$, so the choice of $G$ implies
that every maximal subgroup $M$ of $G$ is $p$-soluble.

(3)  {\sl $F(G)=\Phi (G)$ and $G/F(G)$ is a non-abelian simple group of order divisible by
 $p$. Hence $G'=G$.}

 Let $N$ be any normal proper subgroup of $G$. Then $N$ is $p$-soluble by Claim (2).
Suppose that $N\nleq M$ for some maximal subgroup $M$ of $G$. Then
$G/N\simeq M/M\cap N$ is $p$-soluble by Claim (2) and so $G$ is $p$-soluble,
 contrary to our hypothesis about $G$.
Therefore $N\leq \Phi (G)\leq F(G)$, so $F(G)=\Phi (G)$ and $G/F(G)$ is a
non-abelian simple group of order divisible by $p$.   Hence we have (3).

{\sl The final contradiction.}  Since $G$ is not $p$-soluble, it is not
$p$-nilpotent. Hence $G$ has a subgroup $A$ such that $A$ is not
$p$-nilpotent but every maximal subgroup of $A$ is
$p$-nilpotent. Then $A$ is a $p$-closed Schmidt group with $p\in \pi (A)$
by \cite[Ch. IV,
Satz 5.4]{hupp}.  Then
for some minimal normal subgroup $N$ of $G$ we have $N\leq O_{p}(G)$ by
Claim (1).

 Let  $H/N$ be any  non-subnormal  $p$-subgroup of $G/N$ such that
for some Sylow $p$-subgroup $P/N$ of $G/N$ we have $H/N\leq  (G/N)'\cap
(P/N)$. Then $P$ is a Sylow $p$-subgroup of $G$ and, by Claim (3),  $H$ is a
non-subnormal  $p$-subgroup of $G$ such that  $H\leq P=P\cap G'$.
 Therefore   $H$
  satisfies    the   Frobenius normalizer condition in $G$ by hypothesis and hence
 $H/N$
  satisfies    the   Frobenius normalizer condition in $G/N$ by Lemma 2.3.  Therefore
the hypothesis holds for $G/N$. The choice of $G$ implies
that $G/N$ is $p$-soluble and so  $G$ is $p$-soluble.
The proposition is proved.

We use  in our proof the following properties of the subgroup $Z_{\infty}(G)$.

{\bf Lemma 2.6 }  (See Theorem 2.6 in \cite[Ch. 1]{GuoII}).   {\sl  Let   $Z=Z_{\infty}(G)$.
 Let  $A$, $B$   and $N$ be subgroups of $G$, where
$N$ is normal in $G$.}

(1) {\sl If $ N\leq Z$, then $Z/N= Z_{\infty}(G/N)$.}

(2)   {\sl If $A$ is nilpotent, then $ZA$   is also nilpotent. Hence
$Z$ is contained in each maximal nilpotent subgroup of $G$.}

(3) {\sl If $G/Z$ is nilpotent, then $G$   is also nilpotent.}

  The following Lemma is a corollary of Lemma 2.6(3) and \cite[Ch. A, Theorem 9.3(c)]{DH}.

 {\bf Lemma 2.7.} {\sl  $F(G)/\Phi (G)=F(G/\Phi(G))$ and
$F(G)/Z_{\infty}(G)=F(G/Z_{\infty}(G))$.}

 Let $\phi$ be some linear ordering on  $\mathbb P$.
The record $p\phi q$  means that $p$  precedes $q$ in $\phi$ and $p\ne q$.
Recall that
a group $G$ of order
$p_1^{\alpha _1}p_2^{\alpha _2}\ldots
p_n^{\alpha _n}$
is called \emph{ $\phi$-dispersed} (Baer \cite{baer1}) whenever
$p_{1}\phi p_{2}\phi \ldots \phi p_{n}$
and for every $i$ there is
a normal subgroup of $G$ of order
$p_1^{\alpha _1}p_2^{\alpha _2}\ldots
p_i^{\alpha _i}$.

{\bf Proposition 2.8.} {\sl Suppose that $G$ is soluble.}

(a) {\sl   If every non-normal Sylow subgroup $P$ of $G$ with
 $P\cap G'\ne 1$  satisfies    the   Frobenius normalizer
condition in $G$, then $G$  $\phi$-dispersed for some   linear ordering
$\phi$ on  $\mathbb P$.}

(b)  {\sl If $G$ is not nilpotent but every non-normal Sylow subgroup  of $G$
  satisfies    the   Frobenius
normalizer condition in $G$, then the following conditions holds:}

(i) {\sl  $G$  has a Sylow basis $P_{1}, \ldots ,  P_{t}$
 such that for some $1\leq r < t$ the subgroups
$P_{1}, \ldots ,  P_{r}$ are normal in $G$,  $P_{i}$ is not   normal in
$G$ for all $i > r$
 and $E= P_{r+1} \cdots  P_{t} $  is nilpotent.}

(ii) {\sl $F(G)$ is a maximal
 nilpotent subgroup of $G$  and
  $F(G)=F_{0}(G)Z_{\infty}(G)$, where
 $F_{0}(G)=P_{1} \cdots   P_{r}$.}

(iii) {\sl $N_{G}(E)$ is a Carter subgroup  of $G$.}

(iv) {\sl $V_{G}= Z_{\sigma}(G)$
for every maximal nilpotent subgroup $V$ of $G$ such that  $G=F(G)V$.
}

  {\bf Proof. }  Since $G$ is soluble, it has a Sylow basis $P_{1},
\ldots, P_{t}$.
We can assume without loss of generality that $P_{i}$ is a
$p_{i}$-group for all $i=1, \ldots t.$

 (b)   Assume that this assertion  is false and let $G$ be
 a counterexample of minimal order.

 (1) {\sl  If  $N$ is a  minimal normal subgroup of $G$, then
the  conclusions of the proposition  hold  for
$G/N$.  }

Since $G$ is soluble, $N$ is  primary.
Let $p\in \pi (G/N)$. Suppose that  a Sylow $p$-subgroup $P/N$  of   $G/N$
is not normal  $G/N$. Then $P/N=G_{p}N/N$ for some Sylow $p$-subgroup
$G_{p}$ of $G$. Moreover, \cite[Ch. A, Lemma 14.1(b)]{DH} implies that $G_{p}$
 is not normal in $G$. Hence  $G_{p}$
  satisfies    the   Frobenius condition in $G$ by hypothesis and hence $P/N=G_{p}N/N$
  satisfies    the   Frobenius condition in $G/N$ by Lemma 2.3.
 Therefore the hypothesis holds for $G/N$, so we
 have (1) by the choice of $G$.

(2) {\sl If $P_{i}$ is not normal  in $G$, then $N_{G}(P_{i})$ is $p _{i}$-decomposable. }

Since
$N_{G}(P_{i})/C_{G}(P_{i})$ is a  $p_{i}$-group by hypothesis,  a Hall
 $p_{i}'$-subgroup $V$ of $N_{G}(P_{i})$ is normal in $N_{G}(P_{i})$.
Hence we have (2).

(3) {\sl Statement (b)(i) holds for $G$.}

Since $G$ is not nilpotent, Lemma 2.2 and Claim (2) imply that
   for some $1\leq r < t$ the subgroups
$P_{1}, \ldots ,  P_{r}$ are normal in $G$ and   $P_{i}$ is not   normal in
$G$ for all $i > r$. Then $E= P_{r+1} \cdots  P_{t} $  is nilpotent by Claim (2)
 and Lemma 2.3.  Hence we have  (3).

(4) {\sl Every subgroup $V$
 of $G$ containing $ F(G)$ is subnormal in $G$, so
 $F(V)=F(G)$}.

First note that $G/F(G)=EF(G)/F(G)\simeq E/E\cap F(G)$ is nilpotent by Claim (3).
Therefore $V/F(G)$ is subnormal in $G/F(G)$, which implies that $V$ is subnormal in $G$.
Then  $F(V)\leq F(G)$ by Lemma 2.5. On the other hand, $F(G)\leq F(V)$ and
so   $F(V)=F(G)$.

(5)  {\sl Statement (b)(ii)  holds for $G$.}

First note that  every nilpotent subgroup $V$ of $G$    containing $F(G)$
is subnormal in $G$ by Claim (4), so $V\leq F(G)$ by Lemma 2.5.
Therefore $F(G)$  is a maximal
 nilpotent subgroup of $G$.

In fact, $F(G)=F_{0}(G)\times O_{p_{i_{1}}}(G)\times \cdots \times
 O_{p_{i_{m}}} (G)$ for some $i_{1}, \ldots , i_{m} \subseteq \{r+1,
\ldots , t\}$.
Moreover, in view of Claim (6), we get   that $G/C_{G}(O_{p _{i_{k}}}(G))$  is a
$p_{i_{k}}$-group    for every  $k=1,  \ldots m$ and so  for every chief
factor  $H/K$ of $G$ with   $H\leq O_{p _{i_{k}}}(G)$ we have
$C_{G}(H/K)=G$ since $O_{p _{i_{k}}}(G/C_{G}(H/K))=1$ by \cite[Appendixes, Corollary 6.4]{We}.
 Therefore   $O_{p _{i_{k}}}(G)\leq Z_{\infty}(G)$.
  Hence
  $F(G)=F_{0 }(G)Z_{\infty}(G)$.

(6)  {\sl Statement (b)(iii)  holds for $G$.}

The subgroup $N_{G}(P_{i})$ is $p_{i}$-decomposable for all $i > r$ by Claim
(2). Therefore, by Claim (3),  $$N_{G}(E)=
 N_{G}(P_{r+1}\times \cdots \times P_{t})=N_{G}(P_{r+1})\cap \cdots \cap
N_{G}(P_{t})$$ is nilpotent. On the other hand, $N_{G}(N_{G}(E))=N_{G}(E)$  since $G$ is soluble.
 Hence we have (6).

(7)   Statement (b)(iv)  holds for $G$

First we show that $U_{G}\leq  Z_{\infty}(G)$ for every nilpotent subgroup $U$
 of $G$               such that  $G=F(G)U$.  Suppose that this is false. Then $U_{G}\ne 1$.
 Let $R$ be a minimal
normal subgroup of $G$ contained in $U$ and $C=C_{G}(R)$.
  Then $$G/R=(F(G)R/R)(U/R)=F(G/R)(U/R),$$ so $$U_{G}/R=(U/R)_{G/R}\leq
 Z_{\infty}(G/R)$$ by Claim (1). Since $G$ is soluble, $R$
is a $p$-group for some prime $p$.  Moreover, from $G=F(G)U$
we get that for some Hall $p'$-subgroups $E$, $V$ and
$W$ of $G$, of $F(G)$ and of  $U$, respectively, we have $E=VW$.
But $R\leq F(G)\cap U$, where $F(G)$ and $U$ are
nilpotent.   Therefore $E\leq C$, so $R/1$ is central in
$G$. Hence $R\leq Z_{\infty}(G)$ and so $Z_{\infty}(G/R)=Z_{\infty}(G)/R$ by Lemma 2.6(1).
But then $U_{G}\leq Z_{\infty}(G)$. Finally,  $Z_{\infty}(G)\leq  U$ by
Lemma 2.6(2) and so $U_{G}= Z_{\infty}(G)$.

From Claims (3), (5), (6) and  (7)  it follow that all  conclusions of the
proposition  hold for $G$, contrary to the choice of $G$. This final contradiction completes
 the proof of Assertion (b).

  (a) Assume that this assertion  is false and let $G$ be
 a counterexample of minimal order.

First assume that  that for some $p\in
\pi (G)$  and for a  Sylow $p$-subgroup  $G_{p}$ of we have $G_{p}\cap
G'=1$. Then $G$ is  $p$-nilpotent, so $G$ has a  normal $p$-complement
 $E$. Now let $P$ be a  non-normal Sylow subgroup  of $E$ with
 $P\cap E'\ne 1$. Then $P$ is  a  non-normal Sylow subgroup  of $G$ and
$P\cap G'\ne 1$. Hence $P$ satisfies    the   Frobenius normalizer
condition in $G$ and so   $P$ satisfies    the   Frobenius normalizer
condition in $E$ by Lemma 2.3. Therefore the hypothesis holds for $E$, so
$E$ is $\phi$-dispersed for some   linear ordering
$\phi$
on  $\mathbb P$, so $G$ is $\phi _{0}$-dispersed for some   linear ordering
$\phi _{0}$ on  $\mathbb P$, contrary to our assumption about $G$.
Therefore for every Sylow subgroup $G_{p}$ of $G$ we have $G_{p}\cap G'\ne
1$. But in this case Statement (a) is a corollary of Statement (b).

 The proposition  is proved.

{\bf Proof of Theorem 1.4.}    First note   that  if  every non-subnormal  primary
 subgroup of $G$
 satisfies    the   Frobenius normalizer condition, then in view of Propositions 2.5 and 2.8,
 $G$ is one of the types (a) or (b).

To complete the proof of the theorem it is enough to
  show that if  $G$  is a group of type (b), then  every non-subnormal  primary
 subgroup $H$ of $G$
 satisfies    the   Frobenius condition. Let $H$ be  a $p$-group. Then for
some $i > r$  and $x\in G$ we have $H\leq P_{i}^{x}\leq E^{x}$. Let $N=N_{G}(H)$.
 Then $[N\cap
F_{0}(G), H]=1=[V, H]$, where $V$ is a $p$-complement of  $E^{x}$.
Moreover, $G=F_{0}(G)\rtimes E^{x}$ and so
$N=(F_{0}(G)\cap N)(N\cap E^{x})$, which implies that $N_{G}(H)/C_{G}(H)$ is a
$p$-group.
The theorem is proved.

\section{Proof of Theorem 1.3}

                 \

 Suppose that this theorem  is false and let
$G$ be a counterexample of minimal order.  Then $G$ is not nilpotent.
Moreover, Theorem 1.4 implies that $G$ is soluble and it has a Sylow  basis $P_{1}, \ldots ,  P_{t}$
 such
that  for some $1\leq r < t$ the subgroups
$P_{1}, \ldots ,  P_{r}$ are normal in $G$,  $P_{i}$ is not   normal in
$G$ for all $i > r$
 and $E= P_{r+1} \cdots  P_{t} $  is nilpotent.  Let $F_{0}(G)= P_{1}
\cdots   P_{r}$.

(1) {\sl If $A=A_{p}\rtimes A_{q}$ is any $p$-closed Schmidt
subgroup of $G$,  where $p$ divides $|A|$, then $A_{p}$ is normal in $G$.}

The subgroup
$A_{p}$
 does not satisfy the
 Frobenius normalizer condition in $G$ since $A\leq N_{G}(A_{p}) $   and
$A$ is not $p$-nilpotent. Therefore we have (1) by
hypothesis.

(2) {\sl The  conclusions of the theorem  hold for every proper subgroup  of $G$ and
 for every
quotient $G/N$ of $G$, where $N$ is a minimal normal subgroup of $G$}
 (See the proof of Proposition 2.8 and Theorem 1.4).

(3)  {\sl $G/F(G)$ is abelian. }

It is enough to show that $G'$ is  nilpotent. Suppose
that this is false.  Let $R$ be  a minimal normal subgroup of $G$.

(a) {\sl $R=C_{G}(R)=O_{p}(G)=F(G)\nleq \Phi (G)$ for some prime $p$ and $|R|  > p$}.

 From Claim (2) it follows that for every minimal normal subgroup $N$ of $G$,
 $(G/N)'=G'N/N\simeq G'/G'\cap N$
 is nilpotent.

 If $R\ne N$, it follows that  $G'/((G'\cap N)\cap
(G'\cap R))=G'/1$ is  nilpotent. Therefore $R$ is the  unique
minimal normal subgroup of $G$, $R\leq G'$ and, by Lemma 2.7,  $R\nleq \Phi (G)$. Hence
$R=C_{G}(R)=O_{p}(G)=F(G)$   by    Theorem 15.6 in \cite[Ch. A]{DH}, so
  $|R| > p$ since otherwise
       $G/R=G/C_{G}(R)$ is cyclic, which implies that   $G'=R$ is  nilpotent.

(b)  {\sl $G=P_{1} \rtimes P_{2}$, where $R\leq P_{1}=F(G)$ and $P_{2}$ is
 a minimal non-abelian group.}

From      Claim    (a) it follows that $r=1$  and $R\leq P_{1}=F(G)$.

 Now let
$W=P_{1}V$, where $V$ is a  maximal subgroup of $E$. Then $W$ is
subnormal in $G$ and so $F(W)=F(G)=P_{1}$ (see Claim (7) in the proof of
Theorem 1.4).
But then  $W/P_{1}=P_{1}V/P_{1}\simeq V$ is
abelian by Claim (2). Therefore $E$  is not abelian but
 every proper subgroup of $E$  is abelian, so  $E=P_{2}$ since $E$ is
 nilpotent.  Hence we have (b).

(c)  {\sl $P_{1}=R$  is a Sylow $p$-subgroup of $G$ and every  subgroup $H\ne 1$ of
 $P_{2}$ acts irreducibly  on $R$. Hence every proper  subgroup $H$  of $P_{2}$ is cyclic.}

Since  $P_{1}$  is a normal Sylow $p$-subgroup of $G$,
$P_{1}\leq F(G)\leq C_{G}(R)=R$ by Claim (a)   and  \cite[Ch. A, 13.8(b)]{DH} and  so $P_{1}=R$.

Now let $S=RH$.  By
the Maschke's theorem, $R=R_{1}\times \cdots \times R_{s}$, where $R_{i}$
is a minimal normal subgroup of $S$ for all $s$. Then $R=C_{S}(R)=C_{S}(R_{1}) \cap
\cdots \cap C_{S}(R_{n})$. Hence for some $i$
 the subgroup $R_{i}H$ is not nilpotent and so it has
 a Schmidt
subgroup  $A$ such that $1  < A'$ is normal in $G$ by Claim (1). But then
 $R\leq A$. Therefore $i=1$, so we have (c) since $H$ is abelian by Claim (b).

 {\sl The final contradiction for (3). }   Since every maximal subgroup of  $P_{2}$ is cyclic by Claim (c),
 $q=2$ by
 \cite[Ch. 5,  4.3, 4.4]{Gor}. Therefore $|R|=p$, contrary to
Claim (a).  Hence we have (3).

(4)  {\sl   $G/F(G)$ is cyclic.   }

 Assume that this is false.
First we show that  $Z_{\infty}(G)=1=\Phi (F(G))$.
Assume that for some minimal normal subgroup $R$ of $G$ we have either
$R\leq Z_{\infty}(G)$ or $R\leq  \Phi (F(G))$.
Then $F(G/R)=F(G)/R$ by Lemma 2.7, so Claim (2) implies that
 $(G/R)/F(G/R) =(G/R)/(F(G)/R)  \simeq G/F(G)$ is cyclic.
This contradiction shows that  $Z_{\infty}(G)=1=\Phi (F(G))$.   Therefore
$F(G)=F_{0}(G)$ by Theorem 1.4 and also $F_{0}(G)=R_{1} \times \cdots \times
R_{k}$ for some minimal normal subgroups  $R_{1}, \ldots ,
R_{k}$ of $G$    by \cite[Ch.A, Theorem 10.6(c)]{DH}.
 Since $E\simeq G/F(G)$ is abelian by Claim  (3), and
  $G$ is not nilpotent,  there
 is an index $i$ such that $V=R_{i}\rtimes E$ is not
nilpotent. Then $C_{R_{i}}(E)\ne R_{i}$.   By the Maschke's theorem, $R_{i}=
L_{1}\times \cdots \times L_{m}$
  for some minimal normal subgroups $L_{1}, \ldots ,  L_{m}$ of $V$. Then, since
$C_{R_{i}}(E)\ne R_{i}$,  for some $j$ we have $L_{j}\rtimes E\ne L_{j}\times E$. Hence
$L_{j}E$ contains a Schmidt subgroup $A_{p}\rtimes A_{q}$ such that $A_{p}=R_{i}$ by Claim (1), so $m=1$.
But then  $E$ acts irreducible
on $R_{i}$ and  hence $G/F(G)\simeq E$ is cyclic.  This
contradiction completes the proof of the fact that  $G/F(G)$ is cyclic.

(5) {\sl All maximal nilpotent
subgroups $U$ of $G$ with $F(G)U=G$ are Carter subgroups of $G$.}

 Suppose that this is false.  Assume that for some minimal normal
 subgroup $R$ of $G$ we have
$R\leq Z_{\infty}(G)$.
By Lemma 2.6(2),
  $R\leq  U$. On the other,
$U/R$ is a
maximal nilpotent non-normal subgroup of $G/R$  by
Lemma 2.6(3). Hence     Claim (2) implies that
$U/R$ is a  Carter subgroup  $G/R$,  so
 $U$ is a  Carter subgroup  of $G$.  Hence   $Z_{\infty}(G)=1$, so
    Theorem 1.4 implies that $F(G)=
F_{0}(G)=P_{1}\cdots P_{r}$. Hence $E\simeq  G/F_{0}(G)$ is abelian by Claim (3).
Since   $G=F(G)U $,  for some $x$ we have $E^{x}\leq U$. Hence
$U\leq N_{G}(E^{x})$ since $E^{x}$ is a Hall  subgroup of $G$ and so $U=N_{G}(E^{x})$
  is a
 Carter subgroup of $G$ by    Theorem 1.4(b)(iii).
 This contradiction completes
 the proof of  (5).

Claims (4) and (5) show that the conclusions of the theorem hold for $G$,
 contrary to the choice of $G$.    The theorem is proved.