Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB₁ is a proper subgroup of G, for every proper subgroup B₁ of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary. Supported by BelFBR grant Nos. F05-341 and F06MS-017. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.     

On the behavior of π-submaximal subgroups under homomorphisms

Abstract

We construct examples that show the difference in behavior of π-maximal and π-submaximal subgroups under group homomorphisms.     

On Nearly SS-Embedded Subgroups of Finite Groups

Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H ∩ K ≤ HseG, where HseG is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly SS-embedded subgroups on the structure of finite groups.     

Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations π such that the elements in the closed lower Bruhat interval [id,π] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id,π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner.Our characterisation connects the Poincaré polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups An and Bn. The expressions involve q-Stirling numbers of the second kind, and for the group An putting q=1 yields the poly-Bernoulli numbers defined by Kaneko.     

On the some normal subgroups of the extended modular group

In this paper, we give the group structures and the signatures of some normal subgroups of the extended modular group Π containing the principal congruence subgroup Γ(12).     

Irreducible collineation groups with two orbits forming an oval

Let G be a collineation group of a finite projective plane π of odd order fixing an oval Ω. We investigate the case in which G has even order, has two orbits Ω₀ and Ω₁ on Ω, and the action of G on Ω₀ is primitive. We show that if G is irreducible, then π has a G-invariant desarguesian subplane π₀ and Ω₀ is a conic of π₀.     

Separation of unitary representations of connected Lie groups by their moment sets

We show that every unitary representation π of a connected Lie group G is characterized up to quasi-equivalence by its complete moment set.Moreover, irreducible unitary representations π of G are characterized by their moment sets.     

X-quasinormal subgroups

Considering two subgroups A and B of a group G and ? ≠ X ? G, we say that A is X-permutable with B if AB ^x = B ^x A for some element x ∈ X. We use this concept to give new characterizations of the classes of solvable, supersolvable, and nilpotent finite groups.     

On the <Emphasis Type="Italic">p</Emphasis>-supersolubility of a finite group with a µ-supplemented sylow <Emphasis Type="Italic">p</Emphasis>-subgroup

A subgroup H of a group G is called µ-supplemented in G if there exists a subgroup K such that G = HK and H ₁ K is a proper subgroup in G for every maximal subgroup H ₁ in H. For the initial values of p, we establish the p-supersolubility of a finite group with a μ-supplemented Sylow p-subgroup.     

Conjugacy problem in groups of oriented geometrizable 3-manifolds

The aim of this paper is to show that the fundamental group of an oriented 3-manifold which satisfies Thurston's geometrization conjecture has a solvable conjugacy problem. In other words, for any such 3-manifold M, there exists an algorithm which can decide for any couple of elements u,v of π₁(M) whether u and v are in the same conjugacy class of π₁(M) or not. More topologically, the algorithm decides for any two loops in M, whether they are freely homotopic or not.     

A condition for the solvability of finite groups

A subgroup H is called ?-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H ₁ B is a proper subgroup of G for every maximal subgroup H ₁ of H. We investigate the influence of ?-supplementation of Sylow subgroups and obtain a condition for solvability and p-supersolvability of finite groups.     

Finite groups with <Emphasis Type="Italic">C</Emphasis>-quasinormal subgroups

Consider some finite group G and a finite subgroup H of G. Say that H is c-quasinormal in G if G has a quasinormal subgroup T such that HT = G and T ∩ H is quasinormal in G. Given a noncyclic Sylow subgroup P of G, we fix some subgroup D such that 1 < |D| < | P| and study the structure of G under the assumption that all subgroups H of P of the same order as D, having no supersolvable supplement in G, are c-quasinormal in G.     

Involutory decomposition of a group and twisted subsets with few involutions

A subset K of some group C is called twisted if 1 ∈ K and x, y ∈ K implies that xy ^?1 x belongs to K. We use the concept of twisted subset to investigate and generalize the concept of involutory decomposition of a group. A group is said to admit involutory decomposition if it contains some involution such that the group is the product of the centralizer of the involution and the set of elements inverted by the involution. We study the twisted subsets with at most one involution. We prove that if a twisted subset has no involutions at all then it generates a subgroup of odd order.     

A note on maximal subgroups of free idempotent generated semigroups over bands

We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.     

On an open problem of Guo-Skiba

Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/A _G of G is called a G-boundary factor of A. For any Gboundary factor H/A _G of A, the subgroup (A ∩ H)/A _G of G/ A _G is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p′-group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups.     

Characterization of modular frobenius groups of special type

In this article, we first investigate the properties of modular Frobenius groups. Then, we consider the case that G' is a minimal normal subgroup of a modular Frobenius group G. We give the complete classification of G when G' as a modular Frobenius kernel has no more than four conjugacy classes in G.     

Finite Groups with Few Non-Abelian Subgroups

Let G be a finite group and τ(G) denote the number of conjugacy classes of all non-abelian subgroups of G. The symbol π(G) denotes the set of the prime divisors of |G|. In this paper, finite groups with τ(G) ≤ |π(G)| are classified completely. Furthermore, finite nonsolvable groups with τ(G) = |π(G)| +1 are determined.