Lower bounds on the number of maximal subgroups in a finite group

For a finite group G, let m(G) denote the set of maximal subgroups of G and π(G) denote the set of primes which divide |G|. When G is a cyclic group, an elementary calculation proves that |m(G)| = |π(G)|. In this paper, we prove lower bounds on |m(G)| when G is not cyclic. In general, ${|m(G)| \geq |\pi(G)|+p}$ | m ( G ) | ≥ | π ( G ) | + p , where ${p \in \pi(G)}$ p ∈ π ( G ) is the smallest prime that divides |G|. If G has a noncyclic Sylow subgroup and ${q \in \pi(G)}$ q ∈ π ( G ) is the smallest prime such that ${Q \in {\rm syl}_q(G)}$ Q ∈ syl q ( G ) is noncyclic, then ${|m(G)| \geq |\pi(G)|+q}$ | m ( G ) | ≥ | π ( G ) | + q . Both lower bounds are best possible.     

On the number of maximal subgroups of a finite solvable group

For a finite solvable group G and prime number p, we use elementary methods to obtain an upper bound for \mathfrak m_p(G){\mathfrak {m}_{p}(G)} , defined as the number of maximal subgroups of G whose index in G is a power of p. From this we derive an upper bound on the total number of maximal subgroups of a finite solvable group in terms of its order. This bound improves existing bounds, and we identify conditions on the order of a finite solvable group under which this bound is best possible.     

Influence of properties of maximal subgroups on the structure of a finite group

We establish some tests for the solvability of finite groups and describe one class of unsolvable groups. We prove that an unsolvable group G such that a maximal subgroup M= P × H is nilpotent and the 2-Sylow subgroup P of M is metacyclic has a normal series GD-g₀TD-{1} such that T is contained in M, G₀/T - PSL (2, q), where q is a power of a prime of the form 2ⁿ± 1 and the index of g₀ in G is not greater than 2.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 183–190, February, 1972.The author warmly thanks V. A. Vedernikov for his guidance and assistance in completing this work.     

Generation of a finite group with Hall maximal subgroups by a pair of conjugate elements

For a finite group G, the set of all prime divisors of |G| is denoted by π(G). P. Shumyatsky introduced the following conjecture, which was included in the “Kourovka Notebook” as Question 17.125: a finite group G always contains a pair of conjugate elements a and b such that π(G) = π(〈a, b〉). Denote by \(\mathfrak{Y}\) the class of all finite groups G such that π(H) ≠ π(G) for every maximal subgroup H in G. Shumyatsky’s conjecture is equivalent to the following conjecture: every group from \(\mathfrak{Y}\) is generated by two conjugate elements. Let \(\mathfrak{V}\) be the class of all finite groups in which every maximal subgroup is a Hall subgroup. It is clear that \(\mathfrak{V} \subseteq \mathfrak{Y}\). We prove that every group from \(\mathfrak{V}\) is generated by two conjugate elements. Thus, Shumyatsky’s conjecture is partially supported. In addition, we study some properties of a smallest order counterexample to Shumyatsky’s conjecture.     

On intersections of solvable Hall subgroups in finite nonsolvable groups

The author continues the investigation of intersections of Hall subgroups in finite groups. Previously, the author proved that in the case when a Hall subgroup is Sylow there are three subgroups conjugate to it such that their intersection coincides with the maximal normal primary subgroup. A similar assertion holds for Hall subgroups in solvable groups. The aim of this paper is to construct examples of a (nonsolvable) group in which the intersection of any four subgroups conjugate to some Hall subgroup is nontrivial.     

On finite groups with given maximal subgroups

We prove that a finite group whose every maximal subgroup is simple or nilpotent is a Schmidt group. A group whose every maximal subgroup is simple or supersoluble can be nonsoluble, and in this case we prove that its chief series has the form 1 ? K ? G, K }~ PSL ₂(p) for a suitable prime p, |G: K| ≤ 2.     

On the number of conjugacy classes of maximal subgroups in a finite soluble group

We show that for many formations \frak F\frak F, there exists an integer n = [`(m)](\frak F)n = \overline m(\frak F) such that every finite soluble group G not belonging to the class \frak F\frak F has at most n conjugacy classes of maximal subgroups belonging to the class \frak F\frak F. If \frak F\frak F is a local formation with formation function f, we bound [`(m)](\frak F)\overline m(\frak F) in terms of the [`(m)](f(p))(p ? \Bbb P )\overline m(f(p))(p \in \Bbb P ). In particular, we show that [`(m)](\frak N^k) = k+1\overline m(\frak N^k) = k+1 for every nonnegative integer k, where \frak N^k\frak N^k is the class of all finite groups of Fitting length ￡ k\le k.     

X-permutable maximal subgroups of Sylow subgroups of finite groups

We study finite groups whose maximal subgroups of Sylow subgroups are permutable with maximal subgroups. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1299–1309, October, 2006.     

Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions

We describe the conjugacy classes of finite subgroups in some split extensions using the notion of 1-cocycle and 1-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank 3 is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field.