FINITE GROUPS ALL OF WHOSE SECOND MAXIMAL SUBGROUPS ARE PSC-GROUPS

Generation and random generation: From simple groups to maximal subgroups

The purpose of this paper is to investigate the influence of permutizers of some subgroups of finite groups on the structure of finite groups. We use permutizers of maximal subgroups as well other simple conditions to characterize supersolubility of finite groups. Also we use permutizers of almost maximal subgroups to get a necessary and sufficient condition for a finite group to be supersoluble, which shows that the permutizer condition is, in a sense, equivalent to the condition that PG(M) > M if M is an almost maximal subgroup of G. The paper generalizes and deepens the related results in (1997), Zhang (1986), and Guo (1989).

In [Li and Chen, A new characterization of the simple group [Formula: see text], Sib. Math. J. 53(2) (2012) 213–247.], it is proved that the simple group [Formula: see text] is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as [Formula: see text], Monatsh. Math. 174 (2013) 285–303], the authors proved that if [Formula: see text], where [Formula: see text] is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as [Formula: see text], is isomorphic to [Formula: see text] or an extension of [Formula: see text] by a subgroup of the outer automorphism group of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group with the same orders of maximal abelian subgroups as [Formula: see text], then [Formula: see text] has a unique nonabelian composition factor which is isomorphic to [Formula: see text].

In this paper, we first analyze the structure of a finite nonsolvable group in which every cyclic subgroup of order 2 and 4 of every second maximal subgroup is an NE-subgroup. Next, we prove that a finite group G is solvable if every nonnilpotent subgroup of G is a PE-group.

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

Let G be a finite group with an abelian normal subgroup N. When does N have a unique conjugacy class of complements in G? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups of G closed under conjugation whose intersection equals Φ ⁢ ( G ) {\Phi(G)} . In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when Φ ⁢ ( G ) = 1 {\Phi(G)=1} , these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

Second-metacyclic finite 2-groups are finite 2-groups with some non-metacyclic maximal subgroup and with all second-maximal subgroups being metacyclic. According to a known result there are only four non-metacyclic finite 2-groups with all maximal subgroups being metacyclic. The groups pointed in the title should contain some of these groups as a subgroup of index 2. There are seventeen second-maximal finite 2-groups, four among them being of order 16, ten of order 32 and three of order 64.

Let G be a finite group all of whose proper subgroups are nilpotent. Then by a theorem of Schmidt-Iwasawa the group G is soluble. But what can we say about a finite group G is only one maximal subgroup is nilpotent? Let G be a finite group with a nilpotent maximal subgroup M.

The structure of a finite group having specified number of second and third maximal subgroups has been investigated in the paper.

Let A and G be finite groups such that A acts coprimely on G by automorphisms. We obtain a complete description of the structure of finite groups in which every non-nilpotent maximal A-invariant subgroup of order divisible by p is a TI-subgroup for any fixed prime divisor p of the order of groups.

On the maximal subgroups of finite simple groups

We obtain a full description of nonabelian composition factors for a finite nonsoluble group whose all maximal subgroups are Hall. Thus we solve Problem 17.92 in the "Kourovka Notebook." © 2012 Pleiades Publishing, Ltd.

A proper subgroup [Formula: see text] of a finite group [Formula: see text] is called a second maximal subgroup of [Formula: see text] if [Formula: see text] is a maximal subgroup of every maximal subgroup [Formula: see text] in [Formula: see text] with [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] in which [Formula: see text] is a second maximal subgroup for every non-central element [Formula: see text] in [Formula: see text], and prove that either [Formula: see text] is solvable or [Formula: see text], where [Formula: see text] with [Formula: see text] a prime or [Formula: see text] with [Formula: see text] an odd prime.

Finite simple nonabelian groups G that are not π-closed for some set of primes π but have π-closed maximal subgroups (property (*) for (G, π)) are studied. We give a list L of finite simple groups that contains any group G with the above property (for some π). It is proved that 2 ∉ π for any pair (G, π) with property (*) (Theorem 1). In addition, we specify for any sporadic simple group G from L all sets of primes π such that the pair (G, π) has property (*) (Theorem 2). The proof uses the author’s results on the control of prime spectra of finite simple groups.

Properties of an arbitrary finite π-solvable group whose maximal subgroups are Hall subgroups are established.