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\begin{document}
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\title[To the  Iwasawa and Huppert Theorems \ldots]
{\large To the Iwasawa and Huppert theorems on supersolvability of finite groups}


\author[V.S. Monakhov]{{\bf V.S. Monakhov}}%
\address{Victor Stepanovich Monakhov  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Francisk Skorina Gomel State University,
\newline\hphantom{iii} Kirova Str. 119,
\newline\hphantom{iii} 246019, Gomel, Belarus}%
\email{victor.monakhov@gmail.com}%

\author[I.L. Sokhor]{{\bf I.L. Sokhor}\protect\orcidicon{0000-0002-7115-4045}}
\address{Irina Leonidovna Sokhor  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Brest State A.\,S. Pushkin University,
\newline\hphantom{iii} Kosmonavtov Boulevard 21,
\newline\hphantom{iii} 224016, Brest, Belarus}%
\email{irina.sokhor@gmail.com}%

\thanks{\sc Monakhov, V.S., Sokhor, I.L.
To the theorems of Iwasawa and Huppert}
\thanks{\copyright \ 2024 Monakhov V.S., Sokhor I.L.}
%\thanks{\rm Работа поддержана Белорусским республиканским фондом фундаментальных исследований (грант Ф23РНФ-237)}
\thanks{\rm This work is supported by the Belarusian Republican Foundation for Fundamental Research (Grant $\Phi$23PH$\Phi$-237)}
\thanks{\it Поступила 1 января 2015 г., опубликована 31 декабря 2015 г.}%


\semrtop \vspace{1cm}
\maketitle {\small

\vspace{-12pt}
\centerline{{\it Communicated by} {\sc P.P. Petrov}
%({\it filled by Editorial board})
}

\bigskip
\bigskip

\begin{quote}
\noindent{\bf Abstract:} Let $G\neq 1$ be a finite group and let $\mathbb{P}$ be the set of all primes.
A chain $1=M_0 < M_1< \ldots  < M_{n-1}< M_n=G$ such that
$M_i$ is a maximal subgroup of $M_{i+1}$ for every $i$
is called a maximal chain of~$G$.
Every chain is associated with a sequence of
non-negative integers $j_1$, $j_2$, \ldots, $j_n$, where $j_i=|M_i:M_{i-1}|$.
A maximal chain is a $\mathbb{P}$-chain if $j_i\in\mathbb{P}$ for every $i$.
We say that a $\mathbb{P}$-chain is a $\mathbb{P}^<$-chain ($\mathbb{P}^>$-chain)
if $j_1\leq j_2\leq \ldots\leq j_n$ ($j_1\geq j_2\geq \ldots\geq j_n$,
respectively). We investigate finite groups in which some maximal chains
are $\mathbb{P}$-chains. In particular, we obtain the following  criteria
for finite groups to be supersolvable:
a group~$G$ is supersolvable if and only if
there are a $\mathbb{P}^>$-chain and $\mathbb{P}^<$-chain in $G$;
a group $G$ is supersolvable if and only if
$G$  has a Sylow tower of supersolvable type and
there is a $\mathbb{P}^<$-chain in $G$.
The obtained results are used for characterization
of generally supersolvable groups.
\medskip

\noindent{\bf Keywords:}
finite group, maximal subgroup, chain of subgroups, subgroup index, supersolvable group.
 \end{quote}
}

\bigskip


\section{Introduction}
All groups in this paper are finite.
We use the standard notation and terminology of~\cite{hup}.

Let $G\neq 1$ be a group. A subgroup chain
\begin{equation}\label{e1}
1=M_0\lessdot\,M_1\lessdot\,\ldots \lessdot\,M_i\lessdot\,M_{i+1}\lessdot\,\ldots
\lessdot\,M_{n-1}\lessdot\,M_n=G
\end{equation}
such that $M_i$ is a maximal subgroup of~$M_{i+1}$ for every~$i$
is called a maximal chain of~$G$, and $n$ is the length of this chain.
%Subgroupу $M_{n-k}$, $k\in \mathbb N$,  называют $k$-maximalой subgroupой
%groupы~$G$.
Every chain is associated with a sequence of
non-negative integers $j_1$, $j_2$, \ldots, $j_n$, where $j_i=|M_i:M_{i-1}|$
is the index of $M_{i-1}$ in~$M_i$, $i=1, \ldots, n$.
We use $\mathbb{P}$ to denote the set of all primes.

In 1941, Iwasawa proved the following result.

\begin{theoremnn}[Iwasawa]
A group is supersolvable if and only if its maximal chains have the same length,
\textup{\cite{iwa}}. %{{\cite[VI.9.7]{hup}}}.
\end{theoremnn}

Huppert proved the following fundamental theorem 13 years later.
%фундаментальную

\begin{theoremnn}[Huppert]
A group is supersolvable if and only if all its maximal subgroups are of prime indices,
\textup{\cite[Theorem~9]{h54}}. %{{\cite[VI.9.5]{hup}}}.
\end{theoremnn}

It follows from the Huppert Theorem that all indices of every  maximal chain
in a supersolvable group are primes. Therefore, the following concepts are quite natural.
%Из теоремы Хупперта следует, что последовательности индексов каждой maximalой
%цепи supersolvableой groupы состоят из простых чисел. Поэтому вполне
%естественны следующие понятия.

\begin{definition}
Let~$G$ be a group.  % and ее chain~\eqref{e1} maximalая.
If $j_i\in\mathbb{P}$ for every $i$, then a maximal chain~\eqref{e1}
is called a {\sl $\mathbb{P}$-chain of~$G$.}
We say that a $\mathbb{P}$-chain is a  {\sl $\mathbb{P}^<$-chain}
({\sl $\mathbb{P}^>$-chain})
if $j_1\leq j_2\leq \ldots\leq j_n$ ($j_1\geq j_2\geq \ldots\geq j_n$, respectively).
A maximal chain of~$G$ is said to be a {\sl monotone $\mathbb{P}$-chain}
if it is a $\mathbb{P}^<$-chain or a $\mathbb{P}^>$-chain.
\end{definition}

In this paper, we investigate groups with $\mathbb{P}$-chains.
We enumerate groups in which all maximal chains of every proper subgroup
are monotone $\mathbb{P}$-chains, Theorem~\ref{tmmp}.
In Theorem~\ref{tx}, we indicate the properties of the class
of all groups with $\mathbb{P}^<$-chain.
The following supersolvability criteria follow from this theorem:
{\sl a group $G$ is supersolvable if and only if
there are a $\mathbb{P}^>$-chain and $\mathbb{P}^<$-chain in~$G$;
a group $G$ is supersolvable if and only if
$G$  has a Sylow tower of supersolvable type and
there is a $\mathbb{P}^<$-chain in $G$.}
The obtained results are used for characterization
of generally supersolvable groups, Theorems~\ref{tu},\,\ref{tv}.

%%\newpage

\section{Used notation and concepts.}

If $X$ is a subgroup  (proper subgroup, maximal subgroup)
of a group~$Y$, then we write $X\le Y$
($X<Y$, $X\lessdot \,Y$, respectively).
We use $A\le _tB$ if $A\le B$ and $|B:A|=t$.
We write $r=\max \pi (G)$ ($r=\min \pi (G)$) to
indicate that $r$ is the greatest prime divisor
(the lowest prime divisor, respectively)
of the order of a group~$G$. Here and later,
$\pi(G)$ is the set of all prime divisors of $|G|$.
We use $\mathbb{P}$ and $\mathbb {N}$ to denote the sets of all primes and
all non-negative integers, respectively;
$A_n$ and $S_n$ denotes alternating and symmetric groups of degree~$n$, respectively;
$C_n$ is a cyclic group of order~$n$,
and $C_n^t$ denotes a direct product of $t$ copies of~$C_n$.
We use $A\rtimes B$ to denote the semidirect product
of a normal subgroup $A$ and a subgroup~$B$.

Let $G$ be a group of order $p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_n^{\alpha_n}$,
$p_i\in \mathbb{P}$, $\alpha_i \in \mathbb{N}$, $i=1,\ldots, n$.
We say that  $G$ has {\sl a Sylow tower}
if $G$ has a normal series
\begin{equation}\label{e2}
1 = G_0 < G_1 < \ldots < G_{n-1} < G_n = G
\end{equation}
such that  $|G_i|=p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_i^{\alpha_i}$
for every~$i$. In that case, $G_{i+1}/G_i$ is isomorphic to a Sylow
$p_i$-subgroup of~$G$ for every~$i$.
If $p_1 > p_2 > \ldots  > p_n$, then we say that $G$ has
{\sl a Sylow tower of supersolvable type}, and if
$p_1<p_2< \ldots <p_n$, then $G$ has {\sl a Sylow tower of anti-supersolvable type}.
The classes of all groups with Sylow towers of
supersolvable type and anti-supersolvable type
are denoted by~$\mathfrak{D}$ and~$\overline {\mathfrak{D}}$,
respectively. It is easy to check~$\mathfrak{D}$ and~$\overline {\mathfrak{D}}$
are subgroup-closed hereditary Fitting formations.

Recall a Schmidt group is a non-nilpotent group with all proper subgroups nilpotent.
A group $G$ is a minimal non-supersolvable group if
$G$ is not supersolvable but every proper subgroup of $G$ is supersolvable.
The properties of Schmidt groups  and minimal non-supersolvable groups
are well known, see, for example,~{\color{purple}\cite[p.~26]{shem} or \cite{bsh,bmu}}.
%{\color{purple}A group $G$ is primary if $|G|=p^n$ for a prime $p$.}
A group is primary if it is of prime power order.

\section{Groups with monotone $\mathbb{P}$-chains}

Since composition factors of a solvable group $G\neq 1$
have prime orders, every composition series of a solvable
group is a $\mathbb{P}$-chain.

\begin{example}\label{ex1}
$(1)$~In $A_4$, there is no subgroup of index~$2$,
hence $A_4$ has no $\mathbb{P}^>$-chains, but in $A_4$, there is a $\mathbb{P}^<$-chain:
$1<_2C_2<_2C_2^2<_3A_4$.

$(2)$~In $G=C_3^2\rtimes C_4$ \cite[{SmallGroup(36,9)}]{gap},
there is no subgroup of index~$3$, hence $G$ has no $\mathbb{P}^<$-chains,
but $G$ has a $\mathbb{P}^>$-chain:
$1<_3C_3<_3C_3^2<_2C_3^2\rtimes C_2<_2G$.

$(3)$~In $G=C_3^2\rtimes SL(2,3)$ \cite[{SmallGroup(216,153)}]{gap},
there is no subgroup of index~$2$, hence $G$ has no $\mathbb{P}^>$-chain.
Among the maximal chains of $G$, only three are $\mathbb{P}$-chains:
$$
1<_2 C_2<_3 S_3<_3 C_3\rtimes S_3<_2 C_3^2\rtimes C_4<_2 PSU(3,2)<_3 G,
$$
$$
1<_3 C_3<_2 S_3<_3 C_3\rtimes S_3<_2 C_3^2\rtimes C_4<_2 PSU(3,2)<_3 G,
$$
$$
1<_3 C_3<_3 C_3^2<_2 C_3\rtimes S_3<_2 C_3^2\rtimes C_4<_2 PSU(3,2)<_3 G,
$$
but each of them is not monotone.  In particular,
in~$G=C_3^2\rtimes SL(2,3)$, there is a $\mathbb{P}$-chain,
but there are no $\mathbb{P}^>$-chain and no~$\mathbb{P}^<$-chain.

$(4)$~In~$A_6$, there is no maximal subgroup of prime index,
hence $A_6$ has no $\mathbb{P}$-chain.
\end{example}


\begin{lemma}\label{lpd}
Let $G$ be a group. The following statements hold.

$(1)$~$G$ contains a $\mathbb{P}^>$-chain if and only if
$G$ has a Sylow tower of supersolvable type.

$(2)$~If~$G$ has a Sylow tower of anti-supersolvable type,
then there is a~$\mathbb{P}^{<}$-chain in~$G$.

$(3)$~In any supersolvable group, there is a $\mathbb{P}^<$-chain.
\end{lemma}

\begin{proof}
$(1)$~Assume that in~$G$, there is a $\mathbb{P}^>$-chain~\eqref{e1}.
% and let
%$j_i=|M_i:M_{i-1}|\geq |M_{i+1}:M_i|=j_{i+1}$ for every~$i=1,2,\ldots ,n$.
%Эту chain можно считать произвольной maximalой $>$-chainю.
If $n=1$, then $G$ is a group of prime order and the statement is true.
Therefore we can assume that $n>1$. Use induction on~$n$.
By induction, $M_{n-1}$ has a Sylow tower of supersolvable type.
Hence a Sylow $r$-subgroup $R$ of $M_{n-1}$ is normal in $M_{n-1}$
for $r=j_1=\max \pi (M_{n-1})$. If $r=j_n$, then $G$ is an $r$-group, and
the statement is true. Therefore, $r>j_n$ and $R$ is normal in~$G$
in view of the Sylow Theorem.
By induction, $G/R\in \mathfrak{D}$.
Consequently,  $G\in \mathfrak{D}$.

Conversely, let  $G$ be a group of order
$p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_n^{\alpha_n}$,
$p_1>p_2> \ldots  >p_n$, with a Sylow tower of supersolvable type.
In that case, $G$  has a normal series~\eqref{e2}
such that  $|G_i|=p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_i^{\alpha_i}$
for every~$i$. In particular, $G_1$ is a normal Sylow $p_1$-subgroup
of~$G$ for~$p_1=\max \pi (G)$. We have that $G_1$ contains a chain
$$
1\lessdot \,G_1^{1}\lessdot \, \ldots \lessdot \,G_1^{\alpha_1}=G_1, \
|G_1^{j}:G_1^{j-1}|=p_1, \ j=1,2, \ldots,\alpha_1.
$$
Since $G/G_1$ has a Sylow tower of supersolvable type,
in $G/G_1$ there is a maximal $\mathbb{P}^{>}$-chain
$$
1=G_1/G_1\lessdot \,M_1/G_1\lessdot \ldots \lessdot \,M_m/G_1=G/G_1
$$
by induction.
Now,
$
1\lessdot \,G_1^{1}\lessdot \ldots \lessdot \,G_1^{\alpha_1}
\lessdot \,M_1\lessdot \,M_2\lessdot \ldots \lessdot \,M_m=G
$
is a $\mathbb{P}^{>}$-chain of~$G$.

$(2)$~Use induction on $|G|$.
Let $G\in\overline{\mathfrak{D}}$ and let $r=\max \pi(G)$.
In that case, $G$ has a normal Hall $r'$-subgroup~$H$.
By induction, there is a~$\mathbb{P}^{<}$-chain in $H$.
Continuing this chain to $G$, we obtain a~$\mathbb{P}^{<}$-chain.

$(3)$~Use induction on $|G|$.
Let $G$ be supersolvable and let $r=\max \pi(G)$.
Then $G$ contains a subgroup $H$ of index~$r$.
By induction, there is a $\mathbb{P}^<$-chain in $H$.
Hence in $G$, there is a $\mathbb{P}^<$-chain.
\end{proof}


\begin{example}
In $S_4$, $A_5$, $PSL(2,7)$, there are $\mathbb{P}^{<}$-chains,
but these groups have no Sylow towers.
Therefore, Statement~$(2)$ of Lemma~\ref{lpd} is not converse.
\end{example}

%%\newpage


\begin{theorem}\label{tmmp}
Assume that in all proper subgroups of a group~$G$,
every maximal chain is a monotone $\mathbb{P}$-chain.
Then  $1\leq|\pi (G)|\le 3$ and $G$ is a group
of one of the following type.

$(1)$~$G$ is a supersolvable group of order $p^n$, $pq$, $p^2q$ or~$pqr$;

%$(2)$~$G$~--- $p$-замкнутая groupа порядка~$pq^2$ and $q^2$ делит $(p-1)$;

$(2)$~$G$ is a $p$-closed Schmidt group with Frattini subgroup of prime order~$p$;

$(3)$~$G$ is a non-supersolvable Schmidt group  with identity Frattini subgroup.

Conversely, in all proper subgroups of the enumerated groups,  every maximal chain
is a monotone $\mathbb{P}$-chain.
\end{theorem}

\begin{proof}
Assume that every maximal chain of $G$ is a monotone $\mathbb{P}$-chain.
In that case, all maximal subgroups of $G$ have prime indices,
and $G$ is supersolvable by the Huppert Theorem.
If $G$ is not primary, then in $G$, there is a normal subgroup $C_p$
of prime order~$p$ and a subgroup~$C_q$ of prime order $q$, $p\neq q$.
Suppose that $H=C_pC_q$ is a proper subgroup of~$G$.
If $p>q$, the chain $1\lessdot_q \ C_q\lessdot  _p \ H$
is a $\mathbb{P}^<$-chain and the chain  $1\lessdot_p \ C_p\lessdot_q \ H $
is a $\mathbb{P}^>$-chain.
Fix a monotone  $\mathbb{P}$-chain from~$H$ to~$G$:
$H\lessdot H_1\lessdot\ldots \lessdot H_{n-1}\lessdot H_n=G$.
In that case, one of the  chains
$$
1\lessdot_q C_q\lessdot_p  H\lessdot  H_1\lessdot\ldots\lessdot H_{n-1}\lessdot H_n=G,
$$
$$
1\lessdot  _p \ C_p\lessdot _q \ H\lessdot \ H_1\lessdot \ \ldots \lessdot \ H_{n-1}\lessdot \
H_n=G
$$
are not monotone, a contradiction. Therefore,~$G=C_pC_q$.
Thus, if every maximal chain of~$G$ is monotone $\mathbb{P}$-chain,
then either~$G$ is primary or~$G$ has order~$pq$.

In the sequel, we can assume that $G$ contains a maximal chain
that is not a monotone $\mathbb{P}$-chain.
% немонотонная $\mathbb{P}$-chain.
By the above, every proper subgroup of~$G$ is either primary or of order~$pq$.
Let $G$ be a supersolvable. Then $1\leq |\pi(G)|\leq 3$,
and $|G|=pqr$ if $|\pi(G)|=3$.
If $|\pi(G)|=2$, then $|G|=p^2q$.
Let $G$ be non-supersolvable. In that case,
$G$ is a minimal non-supersolvable group, $G=P\rtimes Q$,
where $Q$ is a supplement to the normal Sylow $p$-subgroup $P$ of~$G$
and $|P/\Phi(P)|=p^n>p$ by~\cite[Lemma~2.1]{mon16}.
If $Q_1\lessdot Q$, $|Q_1|\neq 1$, then $|PQ_1|=p^n|Q_1|\ne pq$, a contradiction.
Consequently, $|Q|=q\in \mathbb{P}$. If $\Phi(P)=1$, then $\Phi(G)=1$ and
$G$ is a Schmidt group with identity Frattini subgroup,
i.\,e. $G$ is a group from Statement~(3) of the theorem.
If $\Phi(P)\neq 1$, then $|\Phi(P)Q|=pq$ and $M=\Phi(P)Q\lessdot G$.
Suppose that $M=\Phi(P)\rtimes Q$ is not nilpotent.
It follows $q$ divides~$p-1$.
In view of $M_G=\Phi(P)$, we have $G/M_G=P/M_G\rtimes M/M_G$
is a Schmidt group with identity Frattini subgroup.
Hence $|G:M|=|P/M_G|=p^n$, where $n$ is the multiplicative order of~$p$ modulo~$q$,
and $n=1$, a contradiction. Hence $M$ is nilpotent, and
$G$ is a $p$-closed Schmidt group with Frattini subgroup of order~$p$,
i.\,e. $G$ is a group from Statement~(2) of the theorem.

It remains to show that in proper subgroups of the enumerated groups,
every maximal chain is a monotone $\mathbb{P}$-chain.
Obviously, the statement is true
for primary groups, groups of order~$pq$,~$p^2q$, or~$pqr$.
If $G$ is a $p$-closed Schmidt group with Frattini subgroup of prime order~$p$,
then $G=P\rtimes Q$, $|P|=p^{1+m}$, $|Q|=q$, where $m$
is the multiplicative order of~$p$ modulo~$q$,
and maximal subgroups of~$G$ have order~$p^{1+m}$ or~$pq$.
It is clear that in all proper subgroups of $G$,
every maximal chain is a monotone $\mathbb{P}$-chain.
In a non-supersolvable Schmidt group with identity Frattini subgroup,
all proper subgroups are primary.
Therefore  such a group also satisfies the conditions of the theorem.
\end{proof}


%%\newpage


\section{On the class of groups with  $\mathbb {P}^<$-chains.}

Let $\mathcal P^<$ ($\mathcal P^>$) be the class of all groups in which there is
a $\mathbb{P}^<$-chain (respectively, $\mathbb{P}^>$-chain).
By $\mathfrak U$ and $\mathfrak S$ we denote the formations of
all supersolvable and all solvable groups, respectively.
In view of Lemma~\ref{lpd}, we have $\mathcal P^>=\mathfrak{D}$,
$\overline{\mathfrak{D}}\subseteq \mathcal P^<$
and~$\mathfrak{U}\subseteq \mathcal P^<$.
In particular, $\mathcal P^>$ is a subgroup-closed hereditary Fitting formation.
The class $\mathcal P^<$ is more complicated.

\begin{theorem}\label{tx}
The following statements hold.

$(1)$~$\mathcal P^<$ is closed under taking normal subgroups, quotients, and direct products.

$(2)$~$\mathcal P^<\cap\mathfrak{S}$ is a subgroup-closed formation.

$(3)$~$\mathcal P^<\cap\mathfrak{D}=\mathcal P^<\cap\mathcal P^>=\mathfrak{U}$.

$(4)$~If a simple non-abelian group $G\in \mathcal P^<$, then
$G\in\{PSL(2,7)$, $PSL(2,11)$, $PSL(2,2^n)\}$,
where  $2^n + 1$ is a Fermat prime.
\end{theorem}

\begin{proof}
$(1)$~Let $G\in\mathcal P^<$ and let $N$ be a normal subgroup of~$G$.
Then in~$G$, there is a $\mathbb P^<$-chain~\eqref{e1}
with prime indices $j_1\leq j_2\leq \ldots\leq j_n$.
Put $K_i=M_i\cap N$ and consider a chain
\begin{equation}\label{peq1}
1=K_0\leq K_1\leq \ldots \leq K_{n-1}\leq K_n=N.
\end{equation}
Since $N$ is normal in $G$, we get $K_{i+1}$ is normal in $M_{i+1}$
and $M_i\leq K_{i+1}M_i\leq M_{i+1}$.
In view of $|M_{i+1}:M_i|=j_{i+1}\in\mathbb{P}$,
either $M_i=K_{i+1}M_i$ or $K_{i+1}M_i=M_{i+1}$.
If $M_i=K_{i+1}M_i$, then $K_i=K_{i+1}$.
If $K_{i+1}M_i= M_{i+1}$, then $j_{i+1}=|M_{i+1}:M_i|=|K_{i+1}M_i:M_i|=|K_{i+1}:K_i|$.
In chain~\eqref{peq1}, we leave only one of the coincide subgroups in chain~\eqref{peq1}
and we get a $\mathcal P^<$-chain of~$N$, i.\,e. $N\in\mathcal P^<$.
Thus,~$\mathcal P^<$ is closed under taking normal subgroups.

It is clear that there is a chain
\begin{equation}\label{peq2}
N/N=M_0N/N\leq \ldots \leq M_{n-1}N/N\leq M_nN/N=G/N.
\end{equation}
Note that
$
|M_{i+1}N/N:M_iN/N|=|M_{i+1}N:M_iN|=|M_{i+1}:M_i|/|(M_{i+1}\cap N):(M_{i}\cap N)|.
$
In view of $|M_{i+1}:M_i|=j_{i+1}\in\mathbb{P}$, we have
either $|(M_{i+1}\cap N):(M_{i}\cap N)|=|M_{i+1}:M_i|$ and $|M_{i+1}N/N:M_iN/N|=1$
or $M_{i+1}\cap N=M_{i}\cap N$ and $|M_{i+1}N/N:M_iN/N|=|M_{i+1}:M_i|=j_{i+1}$.
In chain~\eqref{peq2}, we leave only one of the coincide subgroups in chain~\eqref{peq2}
and we get a $\mathcal P^<$-chain of~$G/N$, i.\,e. $G/N\in\mathcal P^<$.
Thus,~$\mathcal P^<$ is closed under taking quotients.

Now, we check that~$\mathcal P^<$ is closed under taking direct products.
If $A,B\in\mathcal P^<$, then there are $\mathbb{P}^<$-chains
\begin{equation}\label{peq3}
1=A_0<A_1<\ldots < A_{n-1}<A_n=A,
\end{equation}
$$
a_i=|A_i:A_{i-1}|\in \mathbb{P}, \quad
 a_1\leq a_2\leq \ldots\leq a_n,
$$
\begin{equation}\label{peq4}
1=B_0<B_1<\ldots < B_{m-1}<B_m=B,
\end{equation}
$$
b_i=|B_i:B_{i-1}|\in \mathbb{P}, \quad
b_1\leq b_2\leq \ldots\leq b_n.
$$
Use induction on $|A\times B|$.
Let $r=\max\{a_n,b_m\}$.
In that case, in chain~\eqref{peq3},
$A_\alpha$ is a Hall $r'$-subgroup of~$A$
for some~$\alpha$. Similarly, in chain~\eqref{peq4},
$B_\beta$ is a Hall $r'$-subgroup of~$B$
for some~$\beta$.
By induction, $A_\alpha \times B_\beta$ contains a $\mathbb{P}^<$-chain.
It is clear that~$A_\alpha \times B_\beta$ is a Hall $r'$-subgroup
of~$A\times B$, and
we can construct a chain from~$A_\alpha \times B_\beta$ to~$A\times B$
with indices equal to~$r$. So,~$A\times B\in \mathcal P^<$.
%Это означает, что класс~$\mathcal P^<$ замкнут относительно прямых произведений.

$(2)$~Let $G\in \mathcal P^<$ and let $G$ be solvable.
Then in $G$, there is a $\mathbb{P}^<$-chain
$$
1\lessdot M_1\lessdot\ldots\lessdot M_{n-1}=M\lessdot M_n=G.
$$
Hence $|G:M|=r=\max \pi(G)$, and by induction,
there is a $\mathbb{P}^<$-chain in every subgroup of~$M$.
Let $H\lessdot G$. If $H$ and $M$ are conjugate,
then $H$ contains a $\mathbb{P}^<$-chain.
Assume that $H$ and $M$ are not conjugate.
Since $G$ is solvable, we have $G=MH$ and $|G:H|=|H:H\cap M|=r=\max \pi(G)$.
In view of $H\cap M\leq M$, we deduce that $H\cap M$ has a $\mathbb{P}^<$-chain,
and so, $H$ also has a $\mathbb{P}^<$-chain. Thus,
there is a $\mathbb{P}^<$-chain in every maximal subgroup of~$G$.
By induction, in every subgroup of~$G$, there is a $\mathbb{P}^<$-chain,
therefore the class $\mathcal P^<\cap \mathfrak{S}$ is closed under taking subgroups,
i.\,e. it is subgroup-closed.
From Statement~$(1)$, we deduce that $\mathcal P^<\cap \mathfrak{S}$
is a subgroup-closed formation.

$(3)$~By Lemma~\ref{lpd}\,(2-3),
$\mathcal P^<\cap \overline {\mathfrak{D}}=\overline {\mathfrak{D}}$
and~$\mathfrak{U}\subseteq \mathcal P^<$.
Since $\mathfrak{U}\subseteq \mathfrak{D}$,
we get $\mathfrak{U}\subseteq\mathcal P^<\cap \mathfrak{D}$.
Suppose that $\mathcal P^<\cap\mathfrak{D}\nsubseteq\mathfrak{U}$,
and let $G$ be a group of least order such that
$G\in\mathcal P^<\cap\mathfrak{D}\setminus\mathfrak{U}$.
In that case, $G$ is a minimal non-supersolvable group
and $G$ has a Sylow tower of supersolvable type.
Therefore, $|G^\mathfrak{U}/\Phi(G^\mathfrak{U})|=p^n>p$ for
$p=\max \pi(G)$ by~\cite[Lemma~2.1]{mon16}
and $G$ has no subgroup of index~$p$.
Hence $G$ has no $\mathbb P^<$-chain, i.\,e. $G\notin\mathcal P^<$,
a contradiction. Consequently,
$\mathcal P^<\cap\mathfrak{D}\subseteq\mathfrak{U}$ and
$\mathcal P^<\cap\mathfrak{D}=\mathfrak{U}$.

$(4)$~Simple non-abelian groups with $\mathbb{P}$-chains were enumerated
in~\cite[Theorem~3.2]{vvt2010}.
From these groups only $PSL(2,7)$, $PSL(2,11)$ and~$PSL(2,2^n)$,
where  $2^n + 1$ is a Fermat prime, have $\mathbb{P}^<$-chains.
\end{proof}

\begin{corollary}\label{ctx}
A group $G$ is supersolvable if and only if
there are a $\mathbb{P}^{<}$-chain and a $\mathbb{P}^{>}$-chain in $G$.
\end{corollary}

\begin{proof}
If $G$ is supersolvable, then $G$ has a  Sylow tower of supersolvable
type~\cite[VI.9.1(c)]{hup}, and by Lemma~\ref{lpd}\,(1)\,(3),
there are a $\mathbb{P}^>$-chain and a $\mathbb{P}^{<}$-chain in $G$.
Conversely, assume that in~$G$,
there are a $\mathbb{P}^{<}$-chain and a $\mathbb{P}^{>}$-chain.
In that case, $G\in\mathcal P^<$ by the definition of the class~$\mathcal P^<$,
and~$G\in \mathfrak{D}$ by Lemma~\ref{lpd}\,(1).
Thus,~$G\in\mathfrak{D}\cap \mathcal P^<$ and~$G$ is supersolvable
by Theorem~\ref{tx}\,(3).
\end{proof}


\begin{corollary}\label{ctx2}
A group $G$ is supersolvable if and only if
$G$ has a Sylow tower of supersolvable type and
there is a $\mathbb{P}^<$-chain in $G$.
\end{corollary}

\begin{proof}
If $G$ is supersolvable,
then $G$ has a Sylow tower of supersolvable type~\cite[VI.9.1(c)]{hup},
and there is a $\mathbb{P}^<$-chain in $G$ by Lemma~\ref{lpd}\,(3).
Conversely,
if  $G$ has a Sylow tower of supersolvable type~\cite[VI.9.1(c)]{hup}
and there is a $\mathbb{P}^<$-chain in~$G$,
then~$G\in \mathfrak D\cap \mathcal P^<$.
According to Theorem~\ref{tx}\,(3), $G$ is supersolvable.
\end{proof}

\begin{example}\label{exab}
The group $G=C_5^2\rtimes D_8$ \cite[SmallGroup(200,43)]{gap}
contains subgroups~$A$ and~$B$ such that~$A\cong B\cong D_{10}^2$
and~$|G:A|=|G:B|=2$. These subgroups are supersolvable,
therefore they have $\mathbb{P}^<$-chain.
In $G$, there is no maximal subgroups of index~$5$,
hence~$G$ contains no $\mathbb{P}^<$-chain.
Thus, the normal subgroup-closed class $\mathcal P^<$ is not a Fitting class.
\end{example}



%%\newpage

\section{Characterisations of~$\rm{w}$- and~$\rm{v}$-supersolvable groups}

In~\cite{vvt2010}, the following concept was proposed.

\begin{definition}\label{dfpsn}
A subgroup $H$ of a group $G$ is {\sl $\mathbb{P}$-subnormal in}~$G$
if either $G=H$ or there is a chain
\begin{equation}\label{eqpsn}
H=H_0< H_1< \ldots < H_{n-1}< H_n=G
\end{equation}
such that $|H_{i+1}:H_i|\in\mathbb{P}$ for every~$i$.
\end{definition}

It follows from the Huppert Theorem that we can define
the formation~$\mathfrak U$ of all supersolvable groups
as the class of all groups in which every subgroup is $\mathbb{P}$-subnormal.
The classes of all groups with $\mathbb{P}$-subnormal
Sylow subgroups and with $\mathbb{P}$-subnormal
cyclic primary subgroups are denoted by~$\rm{w}\mathfrak U$
and~$\rm{v}\mathfrak U$, respectively.
These classes were quit well investigated~\cite{mon16,vvt2010,mon21,mk13}.
In particular, $\rm{w}\mathfrak U$ and~$\rm{v}\mathfrak U$
are subgroup-closed hereditary formations,
$
\mathfrak U\subset \rm{w}\mathfrak U\subset \rm{v}\mathfrak U
\subset \mathfrak D,
$
and all inclusions are proper.
Groups from~${\rm w}\frak U$ is called {\sl $\mathrm{w}$-supersolvable},
groups from ${\rm v}\frak U$ is called {\sl $\mathrm{v}$-supersolvable}.

In the context of this paper,
the following concepts are quite natural.

\begin{definition}\label{dfpsn+}
A subgroup $H$ of a group~$G$ is {\sl $\mathbb{P}^{<}$-subnormal}
({\sl $\mathbb{P}^{>}$-subnormal}) in~$G$,
if either $G=H$ or there is a chain~\eqref{eqpsn}
such that $|H_{i+1}:H_i|\in\mathbb{P}$ and $|H_i:H_{i-1}|\leq |H_{i+1}:H_i|$
(respectively, $|H_i:H_{i-1}|\ge |H_{i+1}:H_i|$) for every~$i$.
\end{definition}

It is clear that every $\mathbb{P}^{<}$-subnormal and
every $\mathbb{P}^{>}$-subnormal subgroup is $\mathbb{P}$-subnormal.
The converse is not true in general, see Example~\ref{ex1}\,(3).

In this section, supersolvable, $\mathrm{w}$- and $\mathrm{v}$-supersolvable
groups are characterized by the existence or unexistence of $\mathbb{P}^{<}$- or
$\mathbb{P}^{>}$-subnormal subgroups in them.

%%\newpage
\begin{lemma}\label{lp<}
Let $H$ be a $\mathbb{P}^{<}$-subnormal subgroup of a group~$G$,
let $K$ be a subgroup of $G$, and let $N$ be a normal subgroup of~$G$ .
The following statements hold.

$(1)$~$(H \cap N)$ is $\mathbb{P}^{<}$-subnormal in~$N$.

$(2)$~$HN$ is $\mathbb{P}^{<}$-subnormal in~$G$.

$(3)$~$HN/N$ is $\mathbb{P}^{<}$-subnormal in~$G/N$.
\end{lemma}

\begin{proof}
For $\mathbb{P}$-subnormal subgroups, all statements are true, see~\cite[Lemma~1]{mon21}.
By hypothesis, there is a chain~\eqref{eqpsn}
with indices $p_1\le p_2\le \ldots \le p_n$.

$(1)$~It is clear that there is a chain
$$
H\cap N=H_0\cap N\leq H_1\cap N\leq \ldots \leq H_{n-1}\cap N\leq H_n\cap N=N.
$$
Since $N$ is normal in $G$, we deduce that $|H_i\cap N:H_{i-1}\cap N|$ divides
$p_i=|H_i:H_{i-1}|$ for every~$i$. We leave only one of the coincide subgroups
in the chain and we get that $H\cap N$ is $\mathbb{P}^{<}$-subnormal in~$N$.

$(2)$~Since $N$ is normal in $G$, there is a chain
$$
HN=H_0N\leq H_1N\leq \ldots \leq H_{n-1}N\leq H_nN=GN.
$$
In view of
$$
|H_{i+1}N:H_iN|=\frac {|H_{i+1}:H_i|}{|H_{i+1}\cap N:H_i\cap N|}\in
\{1\cup \mathbb P\},
$$
we leave only one of the coincide subgroups in the chain
and we get that $HN$ is $\mathbb{P}^{<}$-subnormal in~$G$.

$(3)$~Since $N$ is normal in $G$, there is a chain
$$
HN/N=H_0N/N\leq H_1N/N\leq \ldots \leq H_{n-1}N/N\leq H_nN/N=GN/N.
$$
In view of
$$
|H_{i+1}N/N:H_iN/N|=\frac {|H_{i+1}:H_i|}{|H_{i+1}\cap N:H_i\cap N|}\in
\{1\cup \mathbb P\},
$$
we leave only one of the coincide subgroups in the chain
and we get that $HN/N$ is $\mathbb{P}^{<}$-subnormal in~$G/N$.
\end{proof}

We can repeat the proof of Lemma~\ref{lp<} with replacing <<$\mathbb{P}^<$-subnormality>>
by <<$\mathbb{P}^>$-subnormality>> to get the similar properties
of~$\mathbb{P}^>$-subnormal subgroups.

\begin{lemma}\label{td>}
The following statements hold.

$(1)$~If $G\in\mathfrak{D}$, then every $\mathbb{P}$-subnormal subgroup of~$G$
is $\mathbb{P}^{>}$-subnormal.

$(2)$~In supersolvable group,
every subgroup is $\mathbb{P}^{>}$-subnormal.

$(3)$~If $G\in\overline{\mathfrak{D}}$,
then every $\mathbb{P}$-subnormal subgroup of~$G$
is $\mathbb{P}^{<}$-subnormal.
\end{lemma}

\begin{proof}
$(1)$~Let $H$ be a $\mathbb{P}$-subnormal subgroup of a group $G\in\mathfrak{D}$.
Use induction on~$|G:H|$.
Assume that $R$ is a Sylow $r$-subgroup of~$G$ for $r=\max \pi(G)$.
In view of~$G\in\mathfrak{D}$, $R$ is normal in~$G$
by the definition of the class~$\mathfrak{D}$.
If $G=HR$, then $H$ is $\mathbb{P}^{>}$-subnormal in $G$.
Assume that $HR$ is a proper subgroup of~$G$.
In view of~\cite[Lemma~4]{mon21}, $HR$ is $\mathbb{P}$-subnormal in~$G$
and~$H$ is $\mathbb{P}$-subnormal in $HR$.
By induction, $HR$ is $\mathbb{P}^{>}$-subnormal in $G$,
hence there is a $\mathbb{P}^{>}$-chain
$$
HR<_{q_1}U_1<_{q_2}U_2<_{q_3} \ldots < _{q_{n-1}}U_{n-1}< _{q_n}U_n=G,
\ \ q_1\ge q_2\ge \ldots \ge q_{n-1}\ge q_n.
$$
In a $\mathbb{P}$-chain from~$H$ to~$HR$, all indices are equal to~$r>q_1$.
Therefore, $H$ is $\mathbb{P}^{>}$-subnormal in~$G$.

$(2)$~It  follows from Statement~$(1)$ because in a supersolvable group,
every subgroup is $\mathbb{P}$-subnormal.

$(3)$~We repeat the proof of Statement~$(1)$ with replacing <<$r=\max \pi(G)$>>
by <<$r=\min\pi(G)$>> and we get that the statement is true.
\end{proof}


\begin{theorem}\label{tu}
Let $G$ be a group. The following statements hold.

$(1)$~If $G$ is supersolvable,
then every Hall subgroup of $G$ is $\mathbb{P}^{>}$-subnormal
and~$\mathbb{P}^{<}$-subnormal in $G$.

$(2)$~$G$ is supersolvable if and only if
every Sylow subgroup of~$G$ is $\mathbb{P}^{<}$-subnormal in~$G$.

$(3)$~$G$ is $\mathrm{w}$-supersolvable if and only if
every Sylow subgroup of $G$ is $\mathbb{P}^{>}$-subnormal in $G$.
\end{theorem}

\begin{proof}
$(1)$~Let $G$ be supersolvable and let $H$ be a Hall subgroup of~$G$.
In view of  Lemma~\ref{td>}\,(2), $H$ is $\mathbb{P}^{>}$-subnormal in~$G$.
Put
$$
\pi =\pi (G)\setminus \pi (H)=\{q_1,q_2,\ldots q_n\}.
$$
Then there is Sylow $q_i$-subgroups~$Q_i$ such that
$HQ_1Q_2\ldots Q_i$ is a subgroup of~$G$ for every~$i$.
Since~$G$ is supersolvable, it follows from the Huppert Theorem that a chain
$$
H<HQ_1<HQ_1Q_2<\ldots  < HQ_1Q_2\ldots Q_{n-1}<G
$$
can be compacted so that all indices of new chain are prime.
It is true for any ordering~$\pi$. For $q_1<q_2<\ldots <q_n$,
a new chain shows that~$H$ is $\mathbb{P}^{<}$-subnormal in~$G$.

$(2)$~If $G$ is supersolvable, then every Sylow subgroup of~$G$
is $\mathbb{P}^{<}$-subnormal in $G$ by Statement~$(1)$.
Conversely, assume that every Sylow subgroup of~$G$ is
$\mathbb{P}^{<}$-subnormal in~$G$.
In that case, $G\in\mathrm{w}\mathfrak{U}\subset\mathfrak{D}$.
In view of Lemma~\ref{lp<}\,(3) and by induction,
$G/N$ is supersolvable for every non-trivial normal subgroup $N$ of~$G$.
Therefore $G$ is a primitive group, and $G=R\rtimes M$,
where $R=F(G)=C_G(R)$ is a Sylow $r$\nobreakdash-subgroup of~$G$
for $r=\max \pi(G)$ and a unique minimal normal subgroup,
and~$M\lessdot \,G$.
Let $Q$ be a Sylow $q$\nobreakdash-subgroup of $M$.
Then $q\neq r$ and $Q$ is a Sylow $q$\nobreakdash-subgroup of~$G$.
By hypothesis, $Q$ is $\mathbb{P}^{<}$\nobreakdash-subnormal in $G$.
Hence, there is a subgroup $H$ of $G$ such that $Q\leq H$ and $|G:H|=r$.
Consequently, $M\leq H^g\leq G$ for some $g\in G$.
Since~$M\lessdot \,G$, we have $M=H^g$ and $|R|=|G:M|=r$.
Therefore, $M\cong G/R=N_G(R)/C_G(R)$ is
a cyclic group of order dividing $r-1$, and $G$ is supersolvable.

%$(2)$~If $G$ is supersolvable, then every Sylow subgroup of~$G$
%is $\mathbb{P}^{<}$-subnormal in $G$ by Statement~$(1)$.
%Conversely, assume that every Sylow subgroup of~$G$ is
%$\mathbb{P}^{<}$-subnormal in~$G$ and $P$ is a Sylow $p$-subgroup of~$G$
%for $p=\min\pi(G)$, $|P|=p^\alpha$. In that case, either $G=H$
%and $G$ is supersolvable or there is a chain
%$$
%P=H_0<_{q_1} H_1<_{q_2} \ldots <_{q_{n-1}} H_{n-1}<_{q_n} H_n=G
%$$
%with prime indices such that $q_1\leq q_2\leq \ldots \leq q_n$.
%In $P$, there is a $\mathbb{P}$-chain with all indices equal to $p<q_1$,
%therefore $G\in\mathcal{P}^<$.
%Since $G$ has a Sylow tower of supersolvable type,
%we deduce that $G\in\mathcal{P}^<\cap\mathfrak{D}$ and  $G$ is supersolvable
%by Theorem~\ref{tx}\,(3).


$(3)$~Since every $\mathrm{w}$-supersolvable group~$G$
has a Sylow tower of supersolvable type, then every Sylow subgroup of $G$
is $\mathbb{P}^{>}$-subnormal in~$G$ by Lemma~\ref{td>}\,(1).
Conversely, if every Sylow subgroup of~$G$ is
$\mathbb{P}^{>}$-subnormal in~$G$, then every Sylow subgroup of~$G$
is $\mathbb{P}$-subnormal in~$G$, and~$G$ is $\mathrm{w}$-supersolvable
by definition.
\end{proof}

%\newpage

\begin{theorem}\label{tv}
Let $G$ be a group. The following statements hold.

$(1)$~If every cyclic primary subgroup of $G$ is $\mathbb{P}^{<}$-subnormal
in~$G$, then $G$ is supersolvable.

$(2)$~$G$ is $\mathrm{v}$-supersolvable if and only if
every cyclic primary subgroup of $G$ is $\mathbb{P}^{>}$-subnormal in $G$.
\end{theorem}

\begin{proof}
$(1)$~Let every cyclic primary subgroup of $G$ be $\mathbb{P}^{<}$-subnormal in~$G$.
In that case, $G\in\mathrm{v}\mathfrak{U}\subset \mathfrak{D}$,
and by Lemma~\ref{lpd}\,(1), $G$ contains a~$\mathbb{P}^{>}$-chain.
Let $q=\min \pi(G)$. By hypothesis, a subgroup of prime order~$q$ is
$\mathbb{P}^{<}$-subnormal in~$G$. Therefore~$G$ contains
a~$\mathbb{P}^{<}$-chain. By Corollary~\ref{ctx}, $G$ is supersolvable.

$(2)$~If~$G\in\mathrm{v}\mathfrak{U}$, then~$G\in\mathfrak{D}$,
and by Lemma~\ref{td>}\,(1), every cyclic primary subgroup is
$\mathbb{P}^{>}$\nobreakdash-subnormal in $G$.
Conversely, if every cyclic primary subgroup $A$ of $G$ is
$\mathbb{P}^{>}$-subnormal in $G$,
then $A$ is $\mathbb{P}$\nobreakdash-subnormal in~$G$,
and~$G\in\mathrm{v}\mathfrak{U}$.
\end{proof}


\bigskip
\begin{thebibliography}{1} % Ссылки на литературу приводятся только на английском языке (даже if у источника нет англоязычной версии).
\bibitem{hup}
B.~Huppert,
    \textit{Endliche Gruppen I},
    Springer, Berlin, Heidelberg, New York, 1967.
    DOI: 10.1007/978-3-642-64981-3

\bibitem{iwa}
K.~Iwasawa,
    \textit{\"{U}ber die endlichen Gruppen und die Verb\"{a}nde ihrer Untergruppen},
    J. Fac. Sci. Imp. Univ. Tokyo. Sect. I, \textbf{4} (1941), 171--199.
    DOI: 10.1007/978-4-431-67947-9\textunderscore 3.


\bibitem{h54}
B.~Huppert,
    \textit{Normalteiler und maximale Untergruppen endlicher Gruppen},
    Math. Z., \textbf{60} (1954), 409--434.
    DOI: 10.1007/BF01187387.

\bibitem{shem}
L.\,A. Shemetkov,
    \textit{Finite groups formations},
    Nauka, Moskow, 1978.
%Эту ссылку оставляем, if статью отправим in русскоязычный журанал. Тогда
%следующие две ссылки убираем. AND наоборот.

\bibitem{bsh}
A.~Ballester-Bolinches, R.~Esteban-Romero, D.J.S.~Robinson,
    \textit{On finite minimal non-nilpotent groups},
    Proc. Am. Math. Soc., \textbf{133}:12 (2005), 3455--3462.
    DOI: 10.1090/S0002-9939-05-07996-7.

\bibitem{bmu}
A.~Ballester-Bolinches, R.~Esteban-Romero,
    \textit{On minimal non-supersoluble groups},
    Rev. Mat. Iberoam., \textbf{23}:1 (2007), 127--142.
    DOI: 10.4171/RMI/488.


\bibitem{gap}
A system for computational discrete algebra GAP 4.12.2 [Electronic resource].
Mode of access: https://www.gap-system.org.

\bibitem{mon16}
V.\,S.~Monakhov,
    \textit{Finite groups with abnormal and $\mathfrak{U}$-subnormal subgroups},
    Sib. Math. J., \textbf{57}:2 (2016) 352--363. DOI: 10.1134/S0037446616020178.

\bibitem{vvt2010}
A.\,F.~Vasil'ev, T.\,I.~Vasilyeva, V.\,N.~Tyutyanov,
    \textit{On finite groups similar to supersoluble groups},
    Probl. Phys. Math. Tech., \textbf{2}(3) (2010), 21--27.

\bibitem{mon21}
V.\,S.~Monakhov,
    \textit{Three Formations over $\mathfrak{U}$},
    Math. Notes, \textbf{110}:3 (2021), 339-346.
    DOI: 10.1134/S0001434621090042.

\bibitem{mk13}
V.\,S.~Monakhov, V.\,N.~Kniahina
    \textit{Finite groups with $\mathbb P$\nobreakdash-subnormal subgroups},
    Ricerche Mat., \textbf{62} (2013), 307--322.
    DOI: 0.1007/s11587-013-0153-9.

\end{thebibliography}
\end{document}
