On the class of finite groups with pronormal Hall subgroups
W. Guo, D. Revin
In this talk, all groups are finite.
We denote by \(\pi\) a set of primes. \(\pi'\) denotes the set of all primes that do not belong
to \(\pi\). For an integer \(n\), \(\pi(n)\) denotes the set of all prime divisors of \(n\). \(\pi(G)\) denotes \(\pi(|G|)\).
Recall that a group \(G\) with \(\pi(G)\subseteq \pi\) is called a
\(\pi\)-group.
A subgroup \(H\) of a group \(G\) is called a \(\pi\)-Hall subgroup if
\(\pi(H)\subseteq\pi\) and \(\pi(|G:H|)\subseteq \pi'\).
According to [1], we say that a group \(G\) satisfies \(\mathscr E_\pi\) if there exists a \(\pi\)-Hall subgroup in \(G\). If
a group \(G\) satisfies \({\mathscr E}_\pi\) and every two \(\pi\)-Hall subgroups of \(G\) are
conjugate in \(G\), then we say that \(G\) satisfies \({\mathscr C}_\pi\). If a group \(G\) satisfies \({\mathscr C}_\pi\) and every
\(\pi\)-subgroup of \(G\) is included in some
\(\pi\)-Hall subgroup of \(G\), then we say that \(G\) satisfies \({\mathscr D}_\pi\). A group satisfying
\({\mathscr E}_\pi\) (resp., \({\mathscr C}_\pi\) and \({\mathscr D}_\pi\)) is
called an \({\mathscr E}_\pi\)-group
(resp., a \({\mathscr C}_\pi\)-group and \({\mathscr D}_\pi\)-group).
The symbols \({\mathscr E}_\pi\),
\({\mathscr C}_\pi\), and \({\mathscr D}_\pi\) denote the classes
of all \({\mathscr E}_\pi\)-, \({\mathscr C}_\pi\)-, and \({\mathscr D}_\pi\)-groups,
respectively.
Recall that, by definition of P. Hall, a
subgroup \(H\) of a group \(G\) is said to be pronormal if the subgroups \(H\) and \(H^g\) are conjugate in \(\langle H, H^g\rangle\) for every \(g\in G\).
As a consequence of the Hall theorem, the Hall subgroups of a solvable
groups are pronormal, whereas there are examples of nonpronormal Hall
subgroups in nonsolvable groups.
We say that a group \(G\) satisfies \({\mathscr P}_\pi\) (is a \({\mathscr P}_\pi\)-group, belongs to the
class \({\mathscr P}_\pi\)) if \(G\in {\mathscr E}_\pi\) and every \(\pi\)-Hall
subgroups in \(G\) is pronormal.
It was established in [2] that the Hall subgroups in every finite simple
group are pronormal. Furthermore, in [3], it was proved that the
\(\pi\)-Hall subgroups are pronormal in every \({\mathscr C}_\pi\)-group. Thus
\[{\mathscr C}_\pi\subseteq {\mathscr P}_\pi\subseteq {\mathscr E}_\pi\]
and every simple \({\mathscr E}_\pi\)-group belongs to \({\mathscr P}_\pi\).
The following problem is well-known: for what
\(\pi\) is the inclusion \({\mathscr C}_\pi\subseteq {\mathscr E}_\pi\) strict? It was established in [4] that
\({\mathscr E}_\pi={\mathscr C}_\pi\) if \(2\notin\pi\), and there are many examples showing that
\({{\mathscr E}_\pi\ne {\mathscr C}_\pi}\), in general.
Also, the following problem is natural: find the sets \(\pi\) for which either of the inclusions
\({{\mathscr C}_\pi\subseteq {\mathscr P}_\pi}\) and \({{\mathscr P}_\pi\subseteq {\mathscr E}_\pi}\) is strict. Firstly, we prove the
following Proposition.
Proposition 1.For every set \(\pi\) of primes, the following
statements are equivalent:
\((1)\)
\({{\mathscr C}_\pi={\mathscr E}_\pi}\);
\((2)\)
\({{\mathscr C}_\pi={\mathscr P}_\pi}\);
\((3)\)
\({{\mathscr P}_\pi={\mathscr E}_\pi}\).
In the theory of classes of finite group and, in particular, in the
theory of Hall's conditions \({\mathscr E}_\pi\), \({\mathscr C}_\pi\), and
\({\mathscr D}_\pi\), the the following operations
play an important rôle, cf. [5]:
\({\rm\small S}{\mathscr X}=\{G\mid G\) is isomorphic to a subgroup of \(H\in {\mathscr X}\}\);
\({\rm\small Q}\mathscr X=\{G\mid G\) is epimorphic image of \(H\in {\mathscr X}\}\);
\({\rm\small S}_n\mathscr X=\{G\mid G\) is isomorphic to a subnormal subgroup of \(H\in {\mathscr X}\}\);
\({\rm\small R}_0\mathscr X=\{G\mid \exists N_i\unlhd G\,\, (i=1,\dots, m)\) with \(G/N_i\in{\mathscr X}\)
and \(\bigcap\limits_{i=1}^m N_i=1 \}\);
\({\rm\small N}_0\mathscr X=\{G\mid \exists N_i{\unlhd\unlhd} G\,\, (i=1,\dots, m) \) with \(N_i\in{\mathscr X}\) and \(G=\langle N_1,\dots, N_m\rangle \}\);
\({\rm\small D}{\mathscr X}=\{G\mid G\simeq H_1\times\dots\times H_m\) for some \(H_i\in {\mathscr X} (i=1,\dots,m)\}\);
\({\rm\small E}\mathscr X=\{G\mid G\) possesses a series \(1=G_0\unlhd G_1\unlhd\dots\unlhd G_m=G \) with \(G_i/G_{i-1}\in {\mathscr X}\) \((i=1,\dots,m)\}\);
\({\rm\small E}_{\rm Z}\mathscr X=\{G\mid \exists N\unlhd G \) with \( N\le{ Z}_\infty(G) \) and \(G/N\in {\mathscr X}\}\);
\({\rm\small E}_{\Phi}\mathscr X=\{G\mid \exists N\unlhd G \) with \( N\le\Phi(G) \) and
\(G/N\in {\mathscr X}\}\);
\({\rm\small P}\mathscr X=\{G\mid \forall H{ < \!\cdot\,} G \,\,G/H_G\in {\mathscr X}\}\).
Here \(H\unlhd\unlhd G\) means that \(H\) is a subnormal subgroup of
\(G\),
and \(H{<\!\cdot\,} G\) means that \(H\) is a maximal subgroup of \(G\). For \(H\le G\), we denote by \(H_G\) the normal subgroup
\(H_G=\bigcap\limits_{g\in G} H^g\). Furthermore, \({ Z}_\infty(G)\) and
\(\Phi(G)\) denote the hypercenter and the Frattini subgroup of a group \(G\), respectively.
For the classes \({\mathscr E}_\pi\), \({\mathscr C}_\pi\), and \({\mathscr D}_\pi\), the
problems of closedness under the operations defined above was in the focus of
attention of many well-known mathematicians during half a century. These
problems play a key role in the theory of these classes (for details, cf. the
surveys [6, 7]).
We investigate the problem of whether or not the class \({\mathscr P}_\pi\) is closed under \({{\rm\small S}}\), \({{\rm\small Q}}\),
\({{\rm\small S}_n}\), \({{\rm\small R}_0}\), \({\rm\small N}_0\),
\({\rm\small D}\), \({\rm\small E}\), \({\rm\small E}_{\rm Z}\),
\({\rm\small E}_{\Phi}\) and \({\rm\small P}\).
The results, together with the analog for the classes \({\mathscr E}_\pi\) and \({\mathscr C}_\pi\), are collected (with the exception of \({\rm\small P}\)) in Table 1.
Table 1. Is it true that
\({\rm\small C}{\mathscr X}={\mathscr X}\), \({{\mathscr X}\in\{{\mathscr E}_\pi, {\mathscr C}_\pi, {\mathscr
P}_\pi\}}\)?
\({\rm\small C}\)
\({\mathscr E}_\pi\)
\({\mathscr C}_\pi\)
\({\mathscr P}_\pi\)
\({\rm\small S}\)
no
no
no
\({\rm\small Q}\)
yes
yes
yes
\({\rm\small S}_n\)
yes
no
no
\({\rm\small R}_0\)
yes
yes
yes
\({\rm\small N}_0\)
no
yes
no
\({\rm\small D}\)
yes
yes
yes
\({\rm\small E}\)
no
yes
no
\({\rm\small E}_{\rm Z}\)
yes
yes
yes
\({\rm\small E}_{\Phi}\)
yes
yes
yes
More precisely, we have the following
Theorem 1.The following statements
hold:
\((A)\)
\({\rm\small C}{\mathscr P}_\pi={\mathscr P}_\pi\) for every set
\(\pi\) of primes and \({\rm\small C}\in \{{{\rm\small Q}},
{{\rm\small R}_0}, {\rm\small D}, {\rm\small E}_{\rm Z},
{\rm\small E}_{\Phi}\}\).
\((B)\)
If \({\rm\small C}\in \{{{\rm\small S}}, {{\rm\small S}_n}, {\rm\small N}_0, {\rm\small E} \}\), then \({\rm\small C}{\mathscr P}_\pi\ne{\mathscr P}_\pi\) for a set \(\pi\) of primes.
\((C)\)
If \({\rm\small C}\in \{{{\rm\small S}_n}, {\rm\small E}\}\) and \({\rm\small C}{\mathscr P}_\pi={\mathscr P}_\pi\) for a set \(\pi\) of primes then \({\rm\small C}{\mathscr E}_\pi={\mathscr E}_\pi\) and \({\rm\small C}{\mathscr C}_\pi={\mathscr C}_\pi\).
Corollary 1.Let \({\rm\small C}\in \{{{\rm\small S}}, {{\rm\small Q}}, {{\rm\small S}_n}, {{\rm\small R}_0}, {\rm\small N}_0, {\rm\small D}, {\rm\small E}, {\rm\small E}_{\rm Z}, {\rm\small E}_{\Phi}\}\).
Then the following statements are equivalent:
\((1)\)
\({\rm\small C}{\mathscr P}_\pi={\mathscr P}_\pi\) for every
set \(\pi\) of primes;
\((2)\)
\({\rm\small C}{\mathscr E}_\pi={\mathscr E}_\pi\) and
\({\rm\small C}{\mathscr C}_\pi={\mathscr C}_\pi\) for every set \(\pi\) of
primes.
Recall that a formation is a \(\langle {\rm\small Q}, {\rm\small R}_0\rangle\)-closed
class \({\mathscr X}\) of finite groups. If, in addition, \({\mathscr X}\) is
\({\rm\small E}_{\Phi}\)-closed then the formation \({\mathscr X}\) is said
to be saturated. A \(\langle {\rm\small Q}, {\rm\small P}\rangle\)-closed class is called Schunk class. Actually,
every saturated formation is a Schunk class.
Corollary 2.For a set \(\pi\) of primes,
\({\mathscr P}_\pi\) is a saturated formation.
Corollary 3.For a set \(\pi\) of primes,
\({\rm\small P}{\mathscr P}_\pi={\mathscr P}_\pi\). In particular,
\({\mathscr P}_\pi\) is a Schunk class.
One can give examples of sets \(\pi\) such that, for every \({\rm\small C}\in \{{{\rm\small S}},
{{\rm\small S}_n}, {\rm\small N}_0, {\rm\small E} \}\), at least one of the classes \({\mathscr E}_\pi\) and
\({\mathscr C}_\pi\) is not closed under \({\rm\small C}\)
(for instance, one can put \(\pi=\{2,3\}\)). It follows from Theorem 1
that, for such a \(\pi\) and every such closure operation
\({\rm\small C}\), the class \({\mathscr P}_\pi\) is not
\({\rm\small C}\)-closed. In particular, \({\mathscr P}_\pi\) is not a Fitting
class (recall, that if
\(\langle{\rm\small S}_n,{\rm\small N}_0\rangle {\mathscr X}={\mathscr X}\) then \({\mathscr X}\) is called a Fitting class).
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