L. A. Shemetkov

ON HALL’S THEOREM

(Presented by Academician A. I. Mal’tsev on 4 VI 1962)

§ 1

The main result of the present note is Theorem 2, which generalizes theorem D5 of Ph. Hall from his paper (¹). The notation used is taken from (¹).

Let \(\Pi\) be some (empty or nonempty) set of primes; \(\mathfrak G\) a finite group of order \(o(\mathfrak G)=g=mn\), where \(m \ge 1\) is the greatest \(\Pi\)-divisor (²) of the order \(g\); a subgroup \(\mathfrak H\) of order \(m\) of the group \(\mathfrak G\) will be called an \(S_{\Pi}\)-subgroup of the group \(\mathfrak G\).

We shall consider the following properties of finite groups: \(E_{\Pi}\)—\(\mathfrak G\) has at least one \(S_{\Pi}\)-subgroup; \(E_{\Pi}^{n}\)—\(\mathfrak G\) has a nilpotent \(S_{\Pi}\)-subgroup; \(C_{\Pi}\)—\(\mathfrak G\) has property \(E_{\Pi}\), and any two \(S_{\Pi}\)-subgroups of the group \(\mathfrak G\) are conjugate in \(\mathfrak G\); \(D_{\Pi}\)—\(\mathfrak G\) has property \(C_{\Pi}\), and every \(\Pi\)-subgroup of the group \(\mathfrak G\) is contained in some \(S_{\Pi}\)-subgroup of the group \(\mathfrak G\); \(D_{\Pi}^{s}\)—\(\mathfrak G\) has property \(D_{\Pi}\), and its \(S_{\Pi}\)-subgroups are solvable.

We now formulate the aforementioned theorem of Ph. Hall.

Theorem 1 ((¹), theorem D5). If \(\mathfrak K\) is such a normal divisor of the group \(\mathfrak G\) that \(\mathfrak K\) has property \(E_{\Pi}^{n}\), and \(\mathfrak G/\mathfrak K\) has property \(D_{\Pi}^{s}\), then \(\mathfrak G\) has property \(D_{\Pi}^{s}\).

§ 2

Definition. A \(\Pi\)-subgroup \(\Omega\) of the group \(\mathfrak G\) will be called a \(\Pi\)-suitable subgroup of the group \(\mathfrak G\) if \(\mathfrak G\) has property \(E_{\Pi}\) and in each conjugacy class of \(S_{\Pi}\)-subgroups of the group \(\mathfrak G\) there is at least one \(S_{\Pi}\)-subgroup containing \(\Omega\).

From this definition the following proposition follows:

A. A finite group has property \(D_{\Pi}\) if and only if all its \(\Pi\)-subgroups are \(\Pi\)-suitable.

Theorem 2. Let \(\mathfrak G\) have an invariant subgroup \(\mathfrak K\) possessing property \(E_{\Pi}^{n}\), and let \(\Omega\) be such a \(\Pi\)-subgroup of the group \(\mathfrak G\) that \(\Omega \mathfrak K/\mathfrak K\) is a solvable \(\Pi\)-suitable subgroup of the group \(\mathfrak G/\mathfrak K\). Then \(\Omega\) is a solvable \(\Pi\)-suitable subgroup of the group \(\mathfrak G\).

Proof. Suppose that the theorem is false. Then choose, among the groups for which the theorem is not satisfied, a group \(\mathfrak G\) of least order. Thus, \(\mathfrak G\) has an invariant subgroup \(\mathfrak K\) possessing property \(E_{\Pi}^{n}\), and in \(\mathfrak G\) there exists such a \(\Pi\)-subgroup \(\Omega\) that \(\Omega \mathfrak K/\mathfrak K\) is a solvable \(\Pi\)-suitable subgroup of the group \(\mathfrak G/\mathfrak K\), but \(\Omega\) is not a solvable \(\Pi\)-suitable subgroup of the group \(\mathfrak G\).

By the hypothesis of the theorem and the definition, \(\mathfrak G/\mathfrak K\) has property \(E_{\Pi}\). By G. Wielandt’s theorem (³), \(\mathfrak K\) has property \(C_{\Pi}\). Hence \(\mathfrak G\) has property \(E_{\Pi}\) by theorem E2 from (¹).

Let \(\mathfrak H\) be an arbitrary \(S_{\Pi}\)-subgroup of the group \(\mathfrak G\). We have to prove that \(\Omega\) is solvable and \(\Omega \subseteq \mathfrak H^{G}\), where \(G \in \mathfrak G\). By lemma 1 from (¹), \(\mathfrak H\mathfrak K/\mathfrak K\) is an \(S_{\Pi}\)-subgroup of the group \(\mathfrak G/\mathfrak K\). Since, by hypothesis, \(\Omega \mathfrak K/\mathfrak K\) is a \(\Pi\)-suitable subgroup of the group \(\mathfrak G/\mathfrak K\), it follows that

\[ \Omega\mathfrak K/\mathfrak K \subseteq (\mathfrak H\mathfrak K/\mathfrak K)^{G_1}=\mathfrak H^{G_1}\mathfrak K/\mathfrak K, \]

where \(G_1 \in \mathfrak G\). Hence it follows that \(\Omega \subseteq \mathfrak H^{G_1}\mathfrak K\). Consider two cases:

1. \((\mathfrak{H}^{G_1}\mathfrak{K}) < (\mathfrak{G})\). Since \(\mathfrak{H}^{G_1}\mathfrak{K}/\mathfrak{K}\) is a \(\Pi\)-subgroup, \(\mathfrak{L}\mathfrak{K}/\mathfrak{K}\) is a solvable \(\Pi\)-suitable subgroup of the group \(\mathfrak{H}^{G_1}\mathfrak{K}/\mathfrak{K}\). Hence, by the induction hypothesis, it follows that \(\mathfrak{L}\) is solvable and \(\mathfrak{L} \subseteq \mathfrak{H}^{G_1G_2}\), where \(G_2 \in \mathfrak{H}^{G_1}\mathfrak{K}\). We have arrived at a contradiction.

2. \(\mathfrak{H}^{G_1}\mathfrak{K} = \mathfrak{G}\). Then also \(\mathfrak{H}^{G_1}\mathfrak{K}\mathfrak{L} = \mathfrak{G}\). Let \(\mathfrak{H}_1 = \mathfrak{H}^{G_1} \cap \mathfrak{K}\mathfrak{L}\). Obviously, the relation
   \[ (\mathfrak{G}) = (\mathfrak{H}^{G_1}\cdot \mathfrak{K}\mathfrak{L}) = \frac{(\mathfrak{H}^{G_1})(\mathfrak{K}\mathfrak{L})}{(\mathfrak{H}_1)} \]
   holds.

It follows from this that \((\mathfrak{G}:\mathfrak{H}^{G_1}) = (\mathfrak{K}\mathfrak{L}:\mathfrak{H}_1)\). Consequently, \(\mathfrak{H}_1\) is an \(S_{\Pi}\)-subgroup of the group \(\mathfrak{K}\mathfrak{L}\). By hypothesis, \(\mathfrak{K}\mathfrak{L}/\mathfrak{K}\) is solvable, and \(\mathfrak{K}\) has property \(E_{\Pi}^{n}\). Hence, by Theorem 1, \(\mathfrak{K}\mathfrak{L}\) has property \(D_{\Pi}^{s}\). Therefore \(\mathfrak{L}\) is solvable and \(\mathfrak{L} \subseteq \mathfrak{H}_1^{G_2}\), where \(G_2 \in \mathfrak{K}\mathfrak{L}\). Since \(\mathfrak{H}_1 \subseteq \mathfrak{H}^{G_1}\), we have
\[ \mathfrak{L} \supseteq \mathfrak{H}_1^{G_2} \subseteq \mathfrak{H}^{G_1G_2}, \]
where \(G_1G_2 \in \mathfrak{G}\). Again we have arrived at a contradiction.

Thus, the supposition of the existence of groups for which the theorem is false always leads to a contradiction. The theorem is thereby proved.

It is easy to see that Theorem 1 follows from Theorem 2, taking into account assumption A.

Theorem 3. Let \(\mathfrak{G}\) have an invariant subgroup \(\mathfrak{K}\) possessing property \(E_{\Pi}^{n}\), and let \(\mathfrak{L}\) be such a \(\Pi\)-subgroup of the group \(\mathfrak{G}\) that \(\mathfrak{L}\mathfrak{K}/\mathfrak{K}\) is solvable and is contained in at least one \(S_{\Pi}\)-subgroup of the group \(\mathfrak{G}/\mathfrak{K}\). Then \(\mathfrak{L}\) is solvable and is contained in at least one \(S_{\Pi}\)-subgroup of the group \(\mathfrak{G}\).

Proof. According to the hypothesis of the theorem,
\[ \mathfrak{L}\mathfrak{K}/\mathfrak{K} \subseteq \mathfrak{H}^*/\mathfrak{K}, \]
where \(\mathfrak{H}^*/\mathfrak{K}\) is some \(S_{\Pi}\)-subgroup of the group \(\mathfrak{G}/\mathfrak{K}\). Hence, \(\mathfrak{L} \subseteq \mathfrak{H}^*\). Since \(\mathfrak{H}^*/\mathfrak{K}\) is a \(\Pi\)-group, \(\mathfrak{L}\mathfrak{K}/\mathfrak{K}\) is a solvable \(\Pi\)-suitable subgroup of the group \(\mathfrak{H}^*/\mathfrak{K}\). Then, by Theorem 2, \(\mathfrak{L}\) is a solvable \(\Pi\)-suitable subgroup of the group \(\mathfrak{H}^*\). Thus, \(\mathfrak{L} \subseteq \mathfrak{H}\), where \(\mathfrak{H}\) is some \(S_{\Pi}\)-subgroup of \(\mathfrak{H}^*\). It is easy to see that \((\mathfrak{H}) = m\), i.e. \(\mathfrak{H}\) is an \(S_{\Pi}\)-subgroup of the group \(\mathfrak{G}\). The theorem is thereby proved.

Gomel Branch
of the Institute of Mathematics and Computer Technology
of the Academy of Sciences of the BSSR

Received
29 V 1962

CITED LITERATURE

² P. Hall, Proc. London Math. Soc., (3) 6, No. 22, 286 (1956).
² S. A. Chunikhin, Matem. sborn., 43(85), No. 1, 49 (1957).
³ H. Wielandt, Math. Zs., 60, 407 (1954).