MATHEMATICS

Ya. G. BERKOVICH

FINITE GROUPS WITH DISPERSIVE SECOND MAXIMAL SUBGROUPS

(Presented by Academician A. I. Mal’tsev, 27 IV 1964)

§ 1. The aim of this paper is a complete description of those nonsolvable groups in which all second maximal subgroups have invariant 2-complements. From this result, in particular, one obtains a description of nonsolvable groups with dispersive second maximal subgroups (here the ordering of the set of primes with respect to which dispersivity is considered is taken in increasing order). Such a description has become possible thanks to the work of Thompson and Feit, who proved the solvability of groups of odd order \((^{1,2})\), of Gorenstein and Walter \((^{3})\) on the structure of groups with a dihedral Sylow 2-subgroup, and also Thompson’s theorem (unpublished), which gives a complete description of minimal simple groups (a simple group is called minimal if all its proper subgroups are solvable)*. More briefly: the aim of the paper is to solve Question 1 posed in § 9 of \((^{15})\).

For the reader’s convenience we state the above-mentioned results of Gorenstein—Walter \((^{3})\) and Thompson.

Theorem A (D. Gorenstein and J. Walter). Let \(G\) be a finite simple group of order \(4g'\) with odd \(g'\), and let its Sylow 2-subgroup coincide with its centralizer. Then \(G \cong LF(2,q)\), where \(q\) is a power of a prime congruent to 3 or 5 modulo 8.

Theorem B (J. Thompson). Let \(G\) be a minimal nonabelian simple group. Then \(G\) is isomorphic to one of the following groups:

(1) \(LF(2,2^p)\), where \(p\) is any prime number.

(2) \(LF(2,3^p)\), where \(p\) is any odd prime number.

(3) \(Sz(2^p)\), where \(p\) is any odd prime number; \(Sz(q)\) is the Suzuki group \((^{4,5})\).

(4) \(LF(2,p)\), where \(p\) is any prime number greater than 3 and such that
\[ p^2 + 1 = 0 \pmod{5}. \]

(5) \(LF(3,3)\).

Only finite groups are considered.

Let us recall some definitions \((^{6})\). We shall say that a group \(G\) has an invariant \(p\)-complement if some homomorphic image of it is isomorphic to a Sylow \(p\)-subgroup of the group (here \(p\) is a prime number). In particular, a group whose order is not divisible by \(p\) also has an invariant \(p\)-complement.

Let \(\varphi\) be some ordering of the set of all prime numbers (all orderings considered here will be total). The ordering \(\varphi\) induces a completely determined ordering on any set of prime numbers (we shall agree to denote this induced ordering by the same letter \(\varphi\)). We shall call a prime number \(p\) \(\varphi\)-minimal divi-

* Thompson’s theorem was formulated in the survey lecture of A. I. Kostrikin at the Fifth All-Union Colloquium on General Algebra in Novosibirsk in 1963.

of the order of the group \(G\), if all divisors of the order of the group \(G\) distinct from \(p\) follow \(p\) in the ordering \(\varphi\). We shall call a group \(G\) \(\varphi\)-dispersive if every subgroup \(H\) of it (\(H\) may coincide with \(G\)) has an invariant \(p\)-complement for the \(\varphi\)-minimal prime divisor \(p\) of the order of \(H\) (cf. with \((^6)\)). The ordering in increasing order will be denoted by the letter \(v\).

§ 2. Theorem 1. Let all second maximal subgroups of a nonsolvable group \(G\) have invariant 2-complements. Then the group \(G\) is isomorphic to one of the following groups:

(1) \(LF(2,2^p)\), where \(2^p-1\) is a Mersenne prime.

(2) \(LF(2,3^p)\), where \(p\) is any odd prime.

(3) \(LF(2,p)\), where \(p\) is any prime \(>3\) such that \(p^2 \equiv 9 \pmod{80}\).

(4) \(SL(2,p)\), where \(p\) is the same prime as in (3).

(5) \(SL(2,3^p)\), where \(p\) is any odd prime.

(6) \(SL(2,5)\).

Proof. Denote by the letter \(\Phi\) the Frattini subgroup of the group \(G\). From the remark on p. 584 of \((^{15})\) it follows that \(G/\Phi\) is a minimal simple group of composite order. Indeed, by Theorem 1 of \((^6)\), the maximal subgroups of the group \(G\) either have invariant 2-complements (and therefore, by \((^1,^2)\), are solvable), or are groups of type \(S\) \((^{7-9})\) (which are also solvable). If \(\Phi=1\), then the result follows immediately from Theorem B (the simple verification that is necessary here we omit). Therefore let \(\Phi\ne1\). Since the property possessed by the group \(G\) is preserved under homomorphism, \(G/\Phi\) is one of the simple groups mentioned in the conclusion of Theorem 1.

Among the proper subgroups of the group \(G\) there is a subgroup \(H\) that has no invariant 2-complement (otherwise \(G\), by \((^6,^1,^2)\), would be solvable). Therefore the subgroup \(H\) is maximal in the group \(G\). It follows that all proper subgroups of the group \(H\) have invariant 2-complements (while \(H\) itself has no such complement); hence \(H\), by Theorem 1 of \((^6)\), is of type \(S\) with an invariant Sylow 2-subgroup. Put \((H)=2^\alpha q^\beta\), where \((H)\) denotes the order of \(H\): \(\alpha,\beta\ge1\); \(q>2\) is a prime (in what follows, the letter \(H\) denotes precisely this subgroup). Consider all possible cases that may arise.

(a) \(G/\Phi \simeq LF(2,2^p)\), \(2^p-1\) is a Mersenne prime.

If \(T\) is a group, then \(T_q\) will denote its Sylow \(q\)-subgroup. Since the group \(G/\Phi_2\) satisfies the condition of the theorem, in order to prove the equality \(\Phi_q=1\) it suffices for the moment to put \(\Phi_2=1\). Since \(G_q=H_q\) and \(H\) is of type \(S\) with a noninvariant Sylow \(q\)-subgroup, \(G_q\) is a cyclic subgroup. As is known from Wielandt’s theorem \((^{10})\), a group with a cyclic Sylow \(q\)-subgroup is either \(q\)-solvable or \(q\)-simple. Since the group \(G\) is not \(q\)-solvable (for even its factor group \(G/\Phi\) is not \(q\)-solvable), it is \(q\)-simple, and therefore \(\Phi_q=1\). Thus \(\Phi\) is a 2-subgroup. If \(p\) is odd, then the theorem of Yu. A. Gol'fand \((^9)\) gives \(a=p\), which is impossible if \(\Phi\ne1\). Therefore \(p=2\). In this case \(q=3\), and the theorem of Yu. A. Gol'fand \((^9)\) gives \(a=3\). Since \(G/\Phi\simeq LF(2,4)\simeq LF(2,5)\) and the order of \(\Phi\) is 2, the application of Schur’s theorem \(((^{11}), p. 120)\) gives \(G\simeq SL(2,5)\).

(b) \(G/\Phi\simeq LF(2,p)\), \(p^2\equiv9\pmod{80}\).

In this case \(H/\Phi\) is a tetrahedral group, and \(q=3\). Again, as in (a), in order to prove the equality \(\Phi_3=1\), suppose for the moment that \(\Phi_2=1\). From the already cited theorem of Wielandt on a group with a cyclic Sylow subgroup \((^{10})\) it follows that \(G_3\) is a noncyclic group. Then in \(G_3\) there is a subgroup \(P\) of order 3 that is not contained in \(\Phi_3\) (\(\Phi_3\) is cyclic, since it is contained in \(H_3\)). But then the subgroup \(P\Phi_3\) is noncyclic. Since in \(G/\Phi_3\) all subgroups of order 3 are conjugate, \(P\Phi_3/\Phi_3\) is conjugate to some subgroup of \(H/\Phi_3\) (denote this subgroup—

than through \(T/\Phi_3\). But then \(P\Phi_3\) and \(T\) are conjugate in \(G\), which is impossible, since \(P\Phi_3\) is noncyclic, while \(T\) is cyclic. Thus, in \(G_3\) there is no subgroup such as \(P\). This is a contradiction, proving the equality \(\Phi_3=1\). Consequently, \(\Phi\) is a 2-subgroup. Since \(q=3\), Yu. A. Gol’fand’s theorem \({}^{(9)}\) gives \(\alpha=3\). In this case \(\Phi\) has order 2. Application of Schur’s theorem \(\left({}^{(11)},\right.\) p. 120) shows that \(G\) is isomorphic to \(SL(2,p)\).

(c) \(G/\Phi \simeq LF(2,3^p)\), where \(p\) is any odd prime. Note that in this case the order of the group \(G/\Phi\) is not divisible by 8.

In this case \(H/\Phi\) is a tetrahedral group. Exactly as in (a) and (b), it is shown that the subgroup \(\Phi\) has order 2 (here a completely analogous method is used, as well as the fact that in \(LF(2,3^p)\) all subgroups of order 3 are conjugate). Since the subgroup \(H_2\) is nonabelian, it contains a cyclic subgroup of order 4 (see \({}^{(9)}\)). Using the method of item (b) and the fact that in \(G/\Phi\) all subgroups of order 2 are conjugate, we shall show that \(\Phi\) is the unique subgroup of order 2 in \(G\) (and therefore also in \(G_2\)). Since the group \(G\) has no invariant 2-complement, its Sylow 2-subgroup \(G_2\) cannot be cyclic. Therefore \(G_2\) is an ordinary quaternion group. It is now clear that \(G\simeq SL(2,3^p)\). The theorem is proved.

Corollary 1. Let \(G\) be a nonsolvable group with \(v\)-dispersive second maximal subgroups. Then it is isomorphic to one of the groups (1)—(4), (6) listed in the conclusion of Theorem 1.

Groups of types (1)—(4), (6) do indeed satisfy the condition of the corollary. A group of type (5), however, does not satisfy the condition of the corollary, since it contains a subgroup of order \(3^p(3^p-1)\), which is not \(v\)-dispersive and is not of type \(S\). But this contradicts Theorem 1 from \({}^{(6)}\).

Now the following is completely obvious.

Corollary 2. Let \(G\) be a nonsolvable group with supersolvable second maximal subgroups. Then it is isomorphic to one of the groups (1), (3), (4), (6) of Theorem 1, and also to the fractional-linear groups \(LF(2,3^p)\) for which the number \(3^p-1\) is equal to twice a prime.

§ 3. Using Theorem A, the Brauer–Suzuki theorem on the nonsimplicity of a group with a quaternion Sylow 2-subgroup \({}^{(14)}\), Theorem 1, and also the methods of \({}^{(15)}\), the following result is proved:

Theorem 2. Let every solvable subgroup \(H\) of a nonsolvable group \(G\) be such that all proper subgroups of \(H\) have invariant 2-complements. Then either \(G\) contains a proper subgroup \(LF(2,2^p)\) \((2^p-1\ge 7\) is a Mersenne prime; in this case the group \(G\) is simple, and \(2^p(2^p-1)\) is the order of the normalizer of its Sylow 2-subgroup), or \(G\) is isomorphic to one of the groups (1)—(6) listed in Theorem 1. Moreover, in (3) and (4) the number \(p\) is congruent to 3 or 5 modulo 8.

We do not know a single example of a group from Theorem 2 that would contain a proper subgroup \(LF(2,2^p)\), where \(2^p-1\ge 7\) is a Mersenne prime. It is clear that Theorem 2 is a generalization of Theorem 1. From Theorem 2 there also follow analogous results of \({}^{(12,13)}\), as well as Theorems 7, 15, 16 of \({}^{(15)}\).

Institute of Mathematics and Computing Technology
Academy of Sciences of the BSSR

Received
15 III 1964

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