The structure of irreducible matrix groups with submultiplicative spectrum

We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p ²× p ² matrices if p is an odd prime.     

The structure of irreducible matrix groups with submultiplicative spectrum

We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p²× p² matrices if p is an odd prime.     

A seminorm with square property is automatically submultiplicative

The result stated in the title is proved in a linear associative algebra, answering a problem posed in [3].     

Finite groups with some CAP-subgroups

Let G be a finite group. A subgroup H of G is called a CAP-subgroup if the following condition is satisfied: for each chief factor K/L of G either HK = HL or H ∩ K = H ∩ L. Let p be a prime factor of |G| and let P be a Sylow p-subgroup of G. If d is the minimum number of generators of P then there exists a family of maximal subgroups of P, denoted by M _d (P)={P ₁, P ₂,…, P _d } such that ∩ _i=1 ^d P _i = ?(P). In this paper, we investigate the group G satisfying the condition: every member of a fixed M _d (P) is a CAP-subgroup of G. For example, if, in addition, G is p-solvable, then G is p-supersolvable.     

Finite groups with transitive semipermutability

A group G is said to be a T-group (resp. PT-group, PST-group), if normality (resp. permutability, S-permutability) is a transitive relation. In this paper, we get the characterization of finite solvable PST-groups. We also give a new characterization of finite solvable PT-groups.      

Finite groups with Frobenius subgroup

Suppose the normalizer N of a subgroup A of a simple group G is a Frobenius group with kernel A, and the intersection of A with any other conjugate subgroup of G is trivial, and suppose, if A is elementary Abelian, that ¦a¦> 2n+1, where n=¦N:A¦. It is proved that if A has a complement B in G, then G acts doubly transitively on the set of right cosets of G modulo B, the subgroup B is maximal in G, and ¦B¦ is divisible by ¦a¦–1. The proof makes essential use of the coherence of a certain set of irreducible characters of N.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 177–186, August, 1976.The author would like to thank V. D. Mazurov for helpful discussions concerning the theorem proved in this note.     

Finite graphs of groups with isomorphic fundamental groups

Translated from Algebra i Logika, Vol. 30, No. 5, pp. 595–623, September–October, 1991.     

Finite groups

The present survey is formed primarily on the basis of works reviewed in the Referativnyi Zhurnal Matematika during 1976–1983 and is a continuation of surveys of the same name published in 1966, 1971, 1976 in the series Albegra, Topologiya, Geometriya (INT). Principal attention is devoted to finite simple groups and their classification.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 24, pp. 3–120, 1986.     

A real seminorm with square property is submultiplicative

A seminorm with square property on a real associative algebra is submultiplicative.     

Finite groups with non-nilpotent maximal subgroups

In this paper, the structure of a finite group under group theoretic restrictions on its non-nilpotent subgroups has been investigated.     

Finite groups with trivial Frattini subgroup

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.