MATHEMATICS

Ya. G. BERKOVICH

A CHARACTERIZATION OF CERTAIN CLASSES OF FINITE GROUPS

(Presented by Academician A. I. Mal’tsev, 25 II 1963)

§ 1. In recent years numerous works have appeared devoted to the characterization of finite groups (see \((^{1-8})\)). The present note is devoted to an analogous question.

Only finite groups are considered. The following notation and concepts are used. \((G)\) is the order of the group \(G\). \(\pi\) is a certain set of primes. \(\pi(G)\) is the set of all prime divisors of \((G)\). \(\pi_G\) is the closure of \(\pi\) in \(G\), i.e. the set containing all prime divisors of each \(\pi d\)-subgroup of the group \(G\). The set \(\pi\) is closed in \(G\) if it coincides with its minimal closure in \(G\) (we note that \(\pi_G\) is not uniquely defined). \((G)_\pi\) is the greatest \(\pi\)-divisor of \((G)\). If

\[ (G)_\pi=\prod_1^k p_i^{\alpha_i} \]

is the canonical decomposition of \((G)_\pi>1\) into prime factors, then

\[ \lambda_\pi(G)=\sum_1^k \alpha_i,\qquad \tau_\pi(G)=k, \]

if \((G)_\pi>1\); if \((G)_\pi=1\), then \(\lambda_\pi(G)=\tau_\pi(G)=0\).

Two groups are called isoordinal if their orders are equal. The totality of subgroups of a group \(G\) that are isoordinal with a given subgroup of it is called a class of isoordinal subgroups. A class of isoordinal subgroups is called noninvariant if it contains at least one noninvariant subgroup \((^{14})\). If a class of isoordinal subgroups contains at least one \((P)\)-subgroup (as groups possessing the group-theoretic property \((P)\) are called), then it is called a class of isoordinal \((P)\)-subgroups.

We shall agree, in the case \(\pi \supseteq \pi(G)\), to omit the symbol \(\pi\) in all notation. \(\overline{S}_k(G)\) is the totality of all those noninvariant soluble subgroups \(H\) of \(G\) for which \(\tau(H)\le k\), \(k\) being a fixed nonnegative integer. \(sn_\pi(G)\) is the number of such classes of isoordinal \(\pi d\)-subgroups of the group \(G\), each of which contains at least one subgroup from \(\overline{S}_3(G)\). \(r_k(G)\) is the number of such classes of isomorphic subgroups of the group \(G\), each of which contains at least one nonnilpotent subgroup from \(\overline{S}_k(G)\). \(N_G(H)\) is the normalizer of the subgroup \(H\) in the group \(G\).

A group of type \(S\) is a nonnilpotent group all of whose proper subgroups are nilpotent \((^{15,16})\). A group \(G\) is called primary if \(\tau(G)=1\). A group of type \(A\) is a nonnilpotent group all of whose proper subgroups are primary \((^{16})\). \(H_G\) is the intersection of all subgroups conjugate to \(H\) in \(G\) \((^{13})\). The definition of the groups \(LF(2,p)\) and \(SL(2,p)\), where \(p\) is a prime number, can be found in \((^2)\).

Subgroups \(H\) and \(F\) form a nilpotent pair if they satisfy at least one of the following conditions:

\[ \text{a) } H\subseteq F;\qquad \text{b) } F\subseteq H;\qquad \text{c) } N_H(H\cap F)\ne H\cap F\ne N_F(H\cap F). \]

A group \(G\) is called quasinilpotent if any two of its subgroups form a nilpotent pair. It is obvious that groups,

all of whose proper subgroups are nilpotent are quasinilpotent.

§ 2. Theorem 1. Let \(G\) be a nonnilpotent quasinilpotent group. Then it has the following properties:

1. \(\tau(G)=2\). Let \(|G|=p^\alpha q^\beta\), and suppose that the subgroup of order \(p^\alpha\) is noninvariant in \(G\).

2. The commutant of the group \(G\) has order \(q^\beta\).

3. The \(p\)-Sylow subgroup of the group \(G\) is cyclic; the subgroups of order \(p^{\alpha-1}\) belong to the center of the group \(G\).

4. All normal divisors of the group \(G\) are nilpotent. All subgroups of the group \(G\) whose order is divisible by \(p^\alpha\) are noninvariant.

5. If \(H\) is a noninvariant maximal subgroup of the group \(G\), then \(G/H_G\) is of type \(A\). All noninvariant maximal subgroups of the group \(G\) are isomorphic and have order \(p^\alpha q^{\beta-b}\), where \(b\) is the least natural number satisfying the congruence \(q^b \equiv 1 \pmod p\).

6. If \(H\) is a noninvariant maximal subgroup of the group \(G\), then \(H=\Phi(G)\), where \(\Phi(G)\) is the Frattini subgroup of the group \(G\). In particular, \(G/\Phi(G)\) is of type \(A\).

7. \(G\) is of type \(S\) if and only if \(\beta<2b\).

It is unknown whether nonnilpotent quasinilpotent groups will always be groups of type \(S\).

Theorem 2. If all nonnilpotent subgroups from \(\overline{S}_3(G)\) are of type \(S\), then only one of the following possibilities occurs:

1. \(G\) is a solvable group.

2. \(G \cong LF(2,5)\).

3. \(G \cong S,L(2,5)\).

Theorem 2 strengthens a result of M. Suzuki \(({}^3)\), p. 695, as well as a theorem of Z. Janko \(({}^{11})\).

Theorem 3. Let \(G\) be a nonsolvable \(\pi d\)-group, \(\pi\) a nonempty set of prime numbers. If

\[ sn_\pi(G)<\lambda_\pi(G)+\tau_\pi(G)+1, \]

then \(G\) is a simple group.

Theorem 4. Let \(G\) be a nonsolvable group and

\[ sn(G)<\lambda(G)+\tau(G)+1. \]

Then either \(G \cong LF(2,5)\), or \(G \cong LF(2,11)\).

Theorem 4 generalizes a result of N. Ito \(({}^9)\).

If we impose no restrictions on the orders of the subgroups under consideration, then instead of the symbols \(sn_\pi(G)\), \(\overline{S}_k(G)\), \(r_k(G)\) we shall write respectively \(s_\pi(G)\), \(\overline{S}(G)\), \(r(G)\). A nonabelian group all of whose maximal subgroups are abelian will be called a group of type \(M\).

Theorem 5. Let \(G\) be a nonsolvable group of even order. If all nonabelian \(2d\)-subgroups from \(\overline{S}(G)\) are of type \(M\), then \(G\) is isomorphic to the icosahedral group.

Theorem 6. Let \(G\) be a nonsolvable group, \(p\) the least prime divisor of \(|G|\), \(\pi_G=\pi\ni p\). If

\[ sn_\pi(G)<2\tau_\pi(G)+2, \]

then either \(G \cong LF(2,5)\), or \(G \cong LF(2,11)\).

Theorem 5 generalizes a result of L. Redei \(({}^6)\), and Theorem 6—a result of N. Ito \(({}^9)\).

Theorem 7. If \(G\) is a nonsolvable group and \(r(G)<\tau(G)+1\), then either \(G \cong LF(2,5)\), or \(G \cong SL(2,5)\).

Theorem 7 generalizes a theorem of D. P. Kolyankovsky \(({}^{17})\). It is of interest to determine whether an analogue of Theorem 7 holds for a theorem of V. I. Sergienko \(({}^{18})\).

Theorem 8. Let \(H \ne 1\) be a solvable normal divisor of a nonsolvable group \(G\). If \(sn(G)<3\tau(G)+2\), then only one of the following possibilities occurs:

1. \(G \cong SL(2,5)\).
2. \(G \cong SL(2,11)\).
3. \(G = G_1 \times H\), where \(G_1 \cong LF(2,5)\), \((H)=2\).

From Theorem 8 follows the result of Ito–Szep \((^{10})\).

Let \(H \ne 1\) be a normal divisor of the group \(G\), and let
\[ G \supset \cdots \supset H \supset H_1 \supset \cdots \supset H_k = 1 \]
be a principal series of the group \(G\) passing through \(H\). Then put \(k=l_G(H)\).

Theorem 9. If \(sn(G)=2\tau(G)+2\), then \(G\) is a solvable group.

Theorem 10. Let \(H \ne 1\) be a solvable normal divisor of a nonsolvable group \(G\). If

\[ s(G)<2\tau(G)+3+l_G(H)[\tau(G)-1], \]

then the group \(G\) has the following properties:

1. \(H\) is a 2-subgroup coinciding with the center of the group \(G\). If \((H)>4\), then \(H=\Phi(G)\).
2. Either \(G/H \cong LF(2,5)\), or \(G/H \cong LF(2,11)\). If \(G/H \cong LF(2,11)\), then \(G \cong SL(2,11)\).

We shall call a subgroup \(H\) of a group \(G\) completely noninvariant if all subgroups of \(G\) that are isomorphic to \(H\) are noninvariant in \(G\).

Theorem 11. In Theorems 3, 4, 6, 8, 9, noninvariance may be replaced by complete noninvariance.

Theorem 12. Let \(P, Q, R\) be pairwise nonisomorphic maximal subgroups of the group \(G\). If \(P, Q, R\) are nilpotent, then \(G\) is nilpotent.

Theorem 13. Let all nonnilpotent maximal subgroups of the group \(G\) be isomorphic. Then \(G\) is solvable and \(\tau(G)\le 3\). If \(\tau(G)=3\), then \(G=P\times H\), where \(H\) is of type \(S\), and \(P\) is a Sylow subgroup of the group \(G\).

Theorem 14. If \(G\) is a nonsolvable group, then it contains at least three noninvariant, pairwise nonisomorphic second maximal subgroups.

Let \(H\) be an arbitrary subgroup of the group \(G\) with \(\lambda(H)=2\). If all maximal chains from \(G\) to \(H\) have one and the same length, then we shall call \(G\) a \(J_2\)-group.

Theorem 15. Let \(G\) be a \(J_2\)-group. Then \(G\) is either supersolvable, or of type \(A\).

Theorem 15 generalizes a known theorem of K. Iwasawa on the supersolvability of \(J\)-groups \((^{12})\).

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

Received
19 II 1963

CITED LITERATURE

\(^{1}\) M. Suzuki, J. Univ. Tokyo, 6, 4, 259 (1951).
\(^{2}\) M. Suzuki, Am. J. Math., 77, 4, 657 (1955).
\(^{3}\) M. Suzuki, Proc. Am. Math. Soc., 8, 4, 686 (1957).
\(^{4}\) M. Suzuki, Trans. Am. Math. Soc., 92, 191 (1959).
\(^{5}\) M. Suzuki, Trans. Am. Math. Soc., 99, 425 (1961).
\(^{6}\) L. Redei, Acta Math., 84, 129 (1950).
\(^{7}\) M. Suzuki, Ann. Math., 75, 105 (1962).
\(^{8}\) W. Feit, Am. J. Math., 82, 281 (1960).
\(^{9}\) N. Ito, Acta Sci. Math., 16, 9 (1955).
\(^{10}\) N. Ito, J. Szep, Acta Sci. Math., 17, 76 (1956).
\(^{11}\) Z. Janko, Math. Zs., 79, 422 (1962).
\(^{12}\) K. Iwasawa, J. Univ. Tokyo, 4—3, 171 (1941).
\(^{13}\) R. Baer, Illinois J. Math., 1, 115 (1957).
\(^{14}\) S. A. Safonov, Scientific Notes of the Belorussian Institute of Engineering Transport, 8, 135 (1958).
\(^{15}\) Yu. I. Smidt, Matematicheskii sbornik, 31, 366 (1924).
\(^{16}\) S. A. Chunikhin, Proceedings of the Seminar on Group Theory, 1938, p. 106.
\(^{17}\) D. P. Kolyankovskii, Matematicheskii sbornik, 19 (61), 429 (1946).
\(^{18}\) V. I. Sergienko, Dokl. AN BSSR, 6, 6, 351 (1962).