MATHEMATICS

S. A. RUSAKOV

THEOREMS OF SYLOW TYPE

(Presented by Academician A. I. Mal'cev on 21 VI 1961)

§ 1. A cycle of papers \((^{1-10})\) contains numerous analogues and generalizations of Sylow’s theorem on the embedding of subgroups in the theory of finite groups. In the article \((^{13})\) (see also \((^{14})\)) we proved two theorems on the embedding of subgroups; moreover, theorem 1 generalizes the known results of G. Wielandt \((^1)\) and Tibiletti \((^9)\).

In the present note we give a further series of theorems obtained by us on the embedding of subgroups, connected with the cyclicity of certain Sylow subgroups of the given group. From theorem 1 of the present note there follows the result of K. Honda \((^{10})\), as well as theorem 2 proved by us earlier \((^{13})\).

§ 2. We shall denote by \(\Pi\) some nonempty set of primes; by \(\Pi(d)\) the totality of all prime divisors of the number \(d\); and by \(\mathfrak G\) a finite group of order \((\mathfrak G)\). Following S. A. Chunikhin \((^{11})\), every divisor \(d\) of the number \((\mathfrak G)\) such that \(\Pi(d)\subseteq \Pi\) will be called a \(\Pi\)-divisor of \((\mathfrak G)\).

Definition 1. We shall call a subgroup of the group \(\mathfrak G\) a \(\Pi\)-subgroup if its order is a \(\Pi\)-divisor of \((\mathfrak G)\).

It is obvious that the identity subgroup of the group \(\mathfrak G\) also belongs to the class of \(\Pi\)-subgroups.

A subgroup whose order is equal to the greatest \(\Pi\)-divisor of \((\mathfrak G)\) will be called, following Wielandt \((^2)\), a \(\Pi\)-Hall subgroup of the group \(\mathfrak G\).

Definition 2. We shall say that for the group \(\mathfrak G\) the strong \(\Pi\)-Sylow theorem holds if, for every \(\Pi\)-subgroup \(\mathfrak H\) of the group \(\mathfrak G\) and for every subgroup \(\mathfrak M\) whose order divides the order of \(\mathfrak H\), there exists an element \(G\in \mathfrak G\) such that
\[ \mathfrak M^G = G^{-1}\mathfrak M G \subseteq \mathfrak H . \]

Definition 3. If there exists a \(\Pi\)-Hall subgroup \(\mathfrak G_\Pi\), and for every subgroup \(\mathfrak M\) whose order divides the order of \(\mathfrak G_\Pi\) there is an embedding
\[ \mathfrak M^G \subseteq \mathfrak G_\Pi,\quad G\in\mathfrak G, \]
then we shall say that for \(\mathfrak G\) the \(\Pi\)-Sylow theorem holds \((^2)\).

In the present paper we also use the concept of a \(\Pi\Delta\)-group, introduced by us in \((^{12})\).

§ 3. Theorem 1. Let to each prime divisor from \(\Pi\) there correspond a cyclic Sylow subgroup of the group \(\mathfrak G\). Then for \(\mathfrak G\) the strong \(\Pi\)-Sylow theorem holds.

Theorem 2. Let \(\Pi\), \(\sigma=\{p\}\), and \(\tau\) be sets of primes such that \(\sigma\cap\tau\) is empty and \(\Pi=\sigma\cup\tau\), and suppose that to the prime number \(p\) there corresponds a cyclic Sylow subgroup of the group \(\mathfrak G\).

Let \(\mathfrak G\) have a \(\Pi\)-subgroup
\[ \mathfrak H=\mathfrak H_p\times \mathfrak G_\tau, \]
where \(\mathfrak H_p\) is a \(p\)-Sylow subgroup of the group \(\mathfrak H\), and \(\mathfrak G_\tau\) is a \(\tau\)-Hall subgroup of the group \(\mathfrak G\).

If for \(\mathfrak G\) the \(\tau\)-Sylow theorem holds, then for any subgroup \(\mathfrak M\) whose order divides the order of \(\mathfrak H\), there exists an element \(G\in\mathfrak G\) such that
\[ \mathfrak M^G \subseteq \mathfrak H . \]

Theorem 3. Let \(\Pi\), \(\sigma\), and \(\tau\) be such (empty or nonempty) sets of prime numbers that \(\sigma\cap\tau\) is empty and \(\Pi=\sigma\cup\tau\), and suppose that to each prime number from \(\sigma\) there corresponds a cyclic Sylow subgroup of the group \(\mathfrak G\).

Let \(G\) have a \(\Pi\)-subgroup \(H=H_\sigma\times G_\tau\), where \(H_\sigma\) is a \(\sigma\)-Hall subgroup of \(H\), and \(G_\tau\) is a \(\tau\)-Hall subgroup of \(G\).

If the \(\tau\)-Sylow theorem holds for \(G\), then for any \(\sigma\)-solvable subgroup \(M\) whose order divides the order of \(H\), there exists an element \(G\in G\) such that \(M^G\subseteq H\).

Theorem 4. Let \(\Pi\), \(\sigma\), and \(\tau\) be such (empty or nonempty) sets of primes that \(\sigma\cap\tau\) is empty, \(\Pi=\sigma\cup\tau\), and every element of \(\sigma\) is smaller than every element of \(\tau\); moreover, to each prime number in \(\sigma\) there corresponds a cyclic Sylow subgroup of the group \(G\).

Let \(G\) have a \(\Pi\)-subgroup \(H=H_\sigma\times G_\tau\), where \(H_\sigma\) is a \(\sigma\)-Hall subgroup of \(H\), and \(G_\tau\) is a \(\tau\)-Hall subgroup of \(G\).

If the \(\tau\)-Sylow theorem holds for \(G\) and \(M\) is any subgroup whose order divides the order of \(H\), then there exists an element \(G\in G\) such that \(M^G\subseteq H\).

Theorem 5. Let \(\Pi\), \(\sigma\), and \(\tau\) be such (empty or nonempty) sets of primes that \(\sigma\cap\tau\) is empty and \(\Pi=\sigma\cup\tau\), and to each prime number in \(\sigma\) there corresponds a cyclic Sylow subgroup of the group \(G\).

Let \(G\) have a \(\Pi\)-subgroup \(H=H_\sigma\times G_\tau\), where \(H_\sigma\) is a \(\sigma\)-Hall subgroup of \(H\), and \(G_\tau\) is a \(\tau\)-Hall subgroup of \(G\).

If the \(\tau\)-Sylow theorem holds for \(G\), then for any \(\sigma\Delta\)-subgroup \(M\) whose order divides the order of \(H\), there exists an element \(G\in G\) such that \(M^G\subseteq H\).

In conclusion I express my sincere gratitude to S. A. Chunikhin for his attention to the work and valuable advice.

Institute of Mathematics and Computational Engineering
Academy of Sciences of the BSSR

Received
11 VI 1961

REFERENCES

\(^{1}\) H. Wielandt, Math. Zs., 60, 407 (1954).
\(^{2}\) H. Wielandt, Math. Zs., 71, 461 (1959).
\(^{3}\) G. Zappa, Boll. Un. Matem. Italiana, 3, Anno 9, No. 4, 349 (1954).
\(^{4}\) S. A. Chunikhin, DAN, 73, 29 (1950).
\(^{5}\) S. A. Chunikhin, Matem. sborn., 39 (81), 465 (1956).
\(^{6}\) S. A. Chunikhin, Matem. sborn., 33 (75), 111 (1953).
\(^{7}\) Ph. Hall, Proc. London Math. Soc., (3), 6, 286 (1956).
\(^{8}\) J. Komaki, Sci. Rep. Tokyo Kyoiku Daigaku, S. A., 6, No. 161, 252 (1959).
\(^{9}\) C. M. Tibiletti, Boll. Un. Matem. Italiana, 3, Anno XIII, No. 1, 46 (1958).
\(^{10}\) K. Honda, Proc. Jap. Acad., 25, No. 1–5, 154 (1949).
\(^{11}\) S. A. Chunikhin, Matem. sborn., 43 (85), 49 (1957).
\(^{12}\) S. A. Rusakov, DAN, 127, No. 2, 270 (1959).
\(^{13}\) S. A. Rusakov, Dokl. AN BSSR, 5, No. 4, 139 (1961).
\(^{14}\) S. A. Rusakov, UMN, 16, issue 2 (98), 223 (1961).