Finite factorizable groups with solvable ℙ2-subnormal subgroups

A subgroup H of a finite group G is called ?²-subnormal whenever there exists a subgroup chain H = H ₀ ≤ H ₁ ≤ ... ≤ H _n = G such that |H _i+1: H _i | divides prime squares for all i. We study a finite group G = AB on assuming that A and B are solvable subgroups and the indices of subgroups in the chains joining A and B with the group divide prime squares. In particular, we prove that a group of this type is solvable without using the classification of finite simple groups.     

On finite groups with generalized σ-subnormal Schmidt subgroups

Let G be a finite group and σ?=?{σ_i|i∈I} some partition of the set of all primes. A subgroup A of G is said to be generalized σ-subnormal in G if A?=??L,T?, where L is a modular subgroup and T is a σ-subnormal subgroup of G. In this paper, we prove that if every Schmidt subgroup of G is generalized σ-subnormal in G, then the commutator subgroup G^′ of G is σ-nilpotent.     

Finite groups with given σ-embedded and σ-n-embedded subgroups

Let G be a finite group and σ = {σ _i |i∈I} be a partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σ _i -subgroup of G and H contains exactly one Hall σ _i -subgroup of G for every σ _i ∈ σ(G). A subgroup H is said to be σ-permutable if G possesses a complete Hall σ-set H such that HA ^x = A ^x H for all A ∈ H and all x ∈ G. Let H be a subgroup of G. Then we say that: (1) H is σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = H ^σG and H ∩ T ≤ H _σG , where H _σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G, and H ^σG is the σ-permutable closure of H, that is, the intersection of all σ-permutable subgroups of G containing H. (2) H is σ-n-embedded in G if there exists a normal subgroup T of G such that HT = H ^G and H ∩ T ≤ H _σG . In this paper, we study the properties of the new embedding subgroups and use them to determine the structure of finite groups.     

Finite groups with ℱ-subnormal conditions

Let ? be a subgroup-closed saturated formation. A finite group G is called an ?_pc-group provided that each subgroup X of G is ?-subabnormal in the ?-subnormal closure of X in G. Let ?_pc be the class of all ?_pc-groups. We study some properties of ? _pc-groups and describe the structure of ?_pc-groups when ? is the class of all soluble π-closed groups, where π is a given nonempty set of prime numbers.     

On K-ℙ-subnormal subgroups of finite groups

A subgroup H of a group G is said to be K-?-subnormal in G if H can be joined to the group by a chain of subgroups each of which is either normal in the next subgroup or of prime index in it. Properties of K-?-subnormal subgroups are obtained. A class of finite groups whose Sylow p-subgroups are K-?-subnormal in G for every p in a given set of primes is studied. Some products of K-?-subnormal subgroups are investigated.     

On the products of ℙ-subnormal subgroups of finite groups

A subgroup H of a finite group G is called ℙ-subnormal in G whenever H either coincides with G or is connected to G by a chain of subgroups of prime indices. If every Sylow subgroup of G is ℙ-subnormal in G then G is called a w-supersoluble group. We obtain some properties of ℙ-subnormal subgroups and the groups that are products of two ℙ-subnormal subgroups, in particular, of ℙ-subnormal w-supersoluble subgroups.     

Finite groups with some maximal subgroups of Sylow subgroups ℳ-supplemented

A subgroup H of a group G is said to be ?-supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H. In this paper, we obtain the following statement: Let ? be a saturated formation containing all supersolvable groups and H be a normal subgroup of G such that G/H ε ?. Suppose that every maximal subgroup of a noncyclic Sylow subgroup of F*(H), having no supersolvable supplement in G, is ?-supplemented in G. Then G ε ?.     

Finite groups with systems of Σ-embedded subgroups

In recent years,a series of papers about cover-avoiding property of subgroups appeared and all the studies were connected with chief factors of a finite group.However,about the cover-avoiding property of subgroups for non-chief factor,there is no study up to now.The purpose of this paper is to build the theory.Let A be a subgroup of a finite group G and Σ:G0≤G1≤…≤Gn some subgroup series of G.Suppose that for each pair(K,H) such that K is a maximal subgroup of H and G i 1 K < H G i for some i,either A ∩ H = ...     

Finite simple groups in which all maximal subgroups are π-closed. I

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.     

Finite factorizable groups with solvable Kℙ<Superscript>2</Superscript>-subnormal subgroups

The solvability of any finite group of the form G = AB is established under the assumption that the subgroups A and B are solvable and KP²-subnormal in the group G.     

Finite simple 3′-groups are cyclic or Suzuki groups

In this note we prove that all finite simple 3′-groups are cyclic of prime order or Suzuki groups. This is well known in the sense that it is mentioned frequently in the literature, often referring to unpublished work of Thompson. Recently an explicit proof was given by Aschbacher [3], as a corollary of the classification of ${\mathcal{S}_3}$ -free fusion systems. We argue differently, following Glauberman’s comment in the preface to the second printing of his booklet [8]. We use a result by Stellmacher (see [12]), and instead of quoting Goldschmidt’s result in its full strength, we give explicit arguments along his ideas in [10] for our special case of 3′-groups.     

Commutator subgroups of the power subgroups of?some Hecke groups

Let q??3 be a prime and let H(?? _q ) be the Hecke group associated to q. Let m be a positive integer and H ^m (?? _q ) be the mth power subgroup of H(?? _q ). In this work, we study the commutator subgroups of the power subgroups H ^m (?? _q ) of H(?? _q ). Then, we give the derived series for all triangle groups of the form (0;2,q,n) for n a positive integer, since there is a nice connection between the signatures of the subgroups we studied and the signatures of these derived series.     

On σ-Embedded and σ-n-Embedded Subgroups of Finite Groups

Siberian Mathematical Journal - Let G be a finite group, and let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and σ(G) = {σi | σi ∩...