MATHEMATICS

V. P. GROMYKO

SOME CRITERIA FOR GENERALIZED SOLVABILITY AND SPECIALNESS OF \(\pi d\)-GROUPS

(Presented by Academician A. I. Mal’cev on 22 IX 1961)

§ 1. The present work is a continuation of our previous investigations of the influence of the number of classes of noninvariant conjugate \(\pi d\)-subgroups of a group \(G\), for a given number of distinct prime \(\pi\)-divisors of the order, on its properties, published in article \((^1)\).

The notation and terminology introduced by us in \((^1)\) are also used in the present note: \(G\) is a finite group of order \(g = mn\), where \(m > 1\) is the largest \(\pi\)-Sylow divisor \((^2)\) of the order \(g\), \(n \geqslant 1\), and, for \(n > 1\),

\[ n = q_1^{\beta_1} q_2^{\beta_2}\ldots q_s^{\beta_s} \]

is the canonical decomposition; a \(\pi d\)-group is a group whose order is divisible by some \(p \in \pi\) \((^3)\); \(t\) is the number of distinct prime \(\pi\)-divisors of the order; \(r\) is the number of classes of noninvariant \(\pi d\)-subgroups of the group \(G\).

Definition 1. Put \(r - t = \lambda\). Then the numbers \(r\) and \(\lambda + 2\) will be called respectively the \(\pi\)-rank and the \(\pi\)-type of the group.

Theorem A (S. A. Chunikhin). If the group \(G\) is \(\pi\)-separable and if \(m\) is such a divisor of its order \(g\) that \(m > 1\) and all prime divisors of \(m\) belong to \(\pi\), and, moreover, \((g/m, m) = 1\), then \(G\) has at least one solvable subgroup of order \(m\), and all subgroups of order \(m\) are conjugate to it \((^4)\).

Definition 2. Let \(m\) be the largest \(\pi\)-Sylow divisor of the order \(g = mn\) of the group \(G\). If \(G\) contains a subgroup \(N\) of order \(n\) and a solvable subgroup \(M\) of order \(m\), and all subgroups of order \(n\) from \(G\) are conjugate to \(N\), while all subgroups of order \(m\) from \(G\) are conjugate to \(M\), then \(G\) will be called a group of type \(\pi - 2\).

Theorem B (S. A. Chunikhin). Every \(\pi\)-solvable group is a group of type \(\pi - 2\) \((^5)\).

Trofimov–Toropov Lemma. If \(G\) is a nonspecial \(\pi d\)-group, then for it \(\lambda \geqslant -1\) \((^{6,7})\).

§ 2. In the present work, with the aid of the theorems of O. Yu. Schmidt \((^8)\), Burnside \((^9)\), and S. A. Chunikhin \((^4)\), as well as the Trofimov–Toropov lemma \((^{6,7})\) and P. I. Trofimov’s “method of intersections” \((^{10}\), Lemmas 1 and \(1'\)), the following main results have been obtained.

Theorem 1. Every \(\pi d\)-group \(G\) of \(\pi\)-type 4, for which \(m \ne p_1^{\alpha_1},\, p_1p_2\) when \(n > 1\), where \(p_1, p_2\) are distinct prime numbers from \(\pi\), is \(\pi\)-solvable.

Theorem 2. Every \(\pi d\)-group \(G\) of \(\pi\)-type 4, for which \(m = p_1^{\alpha_1}\), \(n > 1\), is \(\pi\)-separable. If, however, \(m = p_1p_2\), \(n > 1\), and \(G\) is not \(\pi\)-separable, then it will be a simple group.

From Theorems 1–3 of article \((^1)\) and the theorems formulated here there follow, as corollaries:

Theorem 3. If a \(\pi d\)-group \(G\) is not \(\pi\)-separable, then its \(\pi\)-type is not less than 4.

Theorem 4. Every \(\pi d\)-group \(G\) whose \(\pi\)-type is not greater than 3 is \(\pi\)-solvable, with the exception of the case \(m = p\) for \(\pi\)-type 3. In this exceptional case \(G\) is \(\pi\)-separable.

Theorem 5. Every \(\pi d\)-group \(G\) of \(\pi\)-rank not greater than 4 is \(\pi\)-solvable, with the exception of the cases: \(m=p,\ n>1\), when it is of \(\pi\)-type 3, and \(m=p_1^{\alpha_1},\ p_1p_2\) for \(n>1\), when it is of \(\pi\)-type 4.

Hence, as particular cases, the main results of E. N. Toropov (see \((^{7})\), Theorems 2–6) and the results of O. Yu. Schmidt and P. I. Trofimov \((^{8,11,6})\), concerning the solvability of groups with a number of classes of non-invariant subgroups not greater than 4, are obtained directly.

Theorem 6. Every \(\pi d\)-group \(G\) of \(\pi\)-rank 5 for which \(m\ne p_1^{\alpha_1}p_2^{\alpha_2}\) is \(\pi\)-separable.

Relying on Theorem 6 of \((^{1})\), Theorem 1 of the present paper, and Theorem 2 of S. A. Chunikhin \((^{5})\), we finally obtain the following theorem:

Theorem 7. If \(G\) is a nonspecial \(\pi d\)-group of \(\pi\)-rank greater than 5, then its \(\pi\)-type is not less than 5.

Hence the following corollary follows:

Corollary. \(\pi d\)-groups of \(\pi\)-types 1, 2, 3, and 4 with \(\pi\)-rank greater than 5 are special.

Belorussian Institute
of Railway Transport Engineers

Received
19 IX 1961

REFERENCES

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