Products Of Finite Groups Research Articles

On some products of finite groups

AbstractA classical result of Baer states that a finite group G which is the product of two normal supersoluble subgroups is supersoluble if and only if Gʹ is nilpotent. In this article, we show that if G = AB is the product of supersoluble (respectively, w-supersoluble) subgroups A and B, A is normal in G and B permutes with every maximal subgroup of each Sylow subgroup of A, then G is supersoluble (respectively, w-supersoluble), provided that Gʹ is nilpotent. We also investigate products of subgroups defined above when $ A\cap B=1 $ and obtain more general results.

On $ \sigma $-subnormal subgroups and products of finite groups

<abstract><p>Suppose that $ \sigma = \{ {\sigma}_{i} : i \in I \} $ is a partition of the set $ \mathbb{P} $ of all primes. A subgroup $ A $ of a finite group $ G $ is said to be $ \sigma $-<italic>subnormal</italic> in $ G $ if $ A $ can be joined to $ G $ by a chain of subgroups $ A = A_{0} \subseteq A_{1} \subseteq \cdots \subseteq A_{n} = G $ such that either $ A_{i-1} $ normal in $ A_{i} $ or $ A_{i}/ Core_{A_{i}}(A_{i-1}) $ is a $ {\sigma}_{j} $-group for some $ j \in I $, for every $ 1\leq i \leq n $. A $ \sigma $-subnormality criterion related to products of subgroups of finite $ \sigma $-soluble groups is proved in the paper. As a consequence, a characterisation of the $ \sigma $-Fitting subgroup of a finite $ \sigma $-soluble group naturally emerges.</p></abstract>

Markovian Linearization of Random Walks on Groups

Abstract In operator algebra, the linearization trick is a technique that reduces the study of a non-commutative polynomial evaluated at elements of an algebra ${\mathcal {A}}$ to the study of a polynomial of degree one, evaluated on the enlarged algebra ${\mathcal {A}} \otimes M_r ({\mathbb {C}})$, for some integer $r$. We introduce a new instance of the linearization trick that is tailored to study a finitely supported random walk $G$ by studying instead a nearest-neighbour coloured random walk on $G \times \{1, \ldots , r \}$, which is much simpler to analyze. As an application, we extend well-known results for nearest-neighbour walks on free groups and free products of finite groups to coloured random walks, thus showing how one can obtain new results for finitely supported random walks, namely an explicit description of the harmonic measure and formulas for the entropy and drift.

Normalisers in some products of finite groups

Normalisers in some products of finite groups

On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Szép products

Let L/K be a G-Galois extension of fields with an H-Hopf Galois structure of type N. We study the Galois correspondence ratio GC(G, N), which is the proportion of intermediate fields E with K ⊆ E ⊆ L that are in the image of the Galois correspondence for the H-Hopf Galois structure on L/K. The Galois correspondence ratio for a Hopf Galois structure can be found by translating the problem into counting certain subgroups of the corresponding skew brace. We look at skew braces arising from finite radical algebras A and from Zappa–Sz´ep products of finite groups, and in particular when A3 = 0 or the Zappa–Sz´ep product is a semidirect product, in which cases the corresponding skew brace is a bi-skew brace, that is, a set G with two group operations ◦ and ? in such a way that G is a skew brace with either group structure acting as the additive group of the skew brace. We obtain the Galois correspondence ratio for several examples. In particular, if (G, ◦, ?) is a biskew brace of squarefree order 2m where (G, ◦) ∼= Z2m is cyclic and (G, ?) ∼= Dm is dihedral, then for large m, GC(Z2m, Dm) is close to 1/2 while GC(Dm, Z2m) is near 0.

An Algorithm for Constructing Irreducible Decompositions of Permutation Representations of Wreath Products of Finite Groups

We describe an algorithm for decomposing permutation representations of wreath products of finite groups into irreducible components. The algorithm is based on the construction of a complete set of mutually orthogonal projections to irreducible invariant subspaces of the Hilbert space of the representation under consideration. In constructive models of quantum mechanics, the invariant subspaces of representations of wreath products describe the states of multicomponent quantum systems. The suggested algorithm uses methods of computer algebra and computational group theory. The C implementation of the algorithm is capable of constructing irreducible decompositions of representations of wreath products of high dimensions and ranks. Examples of calculations are given. Bibliography: 15 titles.

On a new type of products of finite groups

On a new type of products of finite groups

Computation of Irreducible Decompositions of Permutation Representations of Wreath Products of Finite Groups

An algorithm for the computation of the complete set of primitive orthogonal idempotents of the centralizer ring of the permutation representation of the wreath product of finite groups is described. This set determines the decomposition of the representation into irreducible components. In the quantum mechanics formalism, the primitive idempotents are projection operators onto irreducible invariant subspaces of the Hilbert space describing the states of many-particle quantum systems. The proposed algorithm uses methods of computer algebra and computational group theory. The C implementation of this algorithm is able to construct decompositions of representations of high dimensions and ranks into irreducible components.

Group algebras of free products of finite groups which are CS-rings

This paper proves that the infinite dihedral group is only such a free product of two finite groups that its group algebra over a field [Formula: see text] is a CS-ring in case the orders of two groups are not zero in [Formula: see text]. Furthermore, it is shown that the group algebra of any free product of two finite cyclic groups does not satisfy the condition [Formula: see text].

Some remarks on finitarily approximable groups

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite solvable groups with arbitrary invariant length function. We answer this question by showing that any non-trivial finitely generated perfect group does not have this property, generalizing a counterexample of Howie. Moreover, we discuss the question which connected Lie groups can be embedded into a metric ultraproduct of finite groups with invariant length function. We prove that these are precisely the abelian ones, providing a negative answer to a question of Doucha. Referring to a problem of Zilber, we show that a the identity component of a Lie group, whose topology is generated by an invariant length function and which is an abstract quotient of a product of finite groups, has to be abelian. Both of these last two facts give an alternative proof of a result of Turing. Finally, we solve a conjecture of Pillay by proving that the identity component of a compactification of a pseudofinite group must be abelian as well. All results of this article are applications of theorems on generators and commutators in finite groups by the first author and Segal.

Influence of minimal subgroups on the product of smooth groups

A maximal chain in a finite lattice L is called smooth if any two intervals of the same length are isomorphic. We say that a finite group G is totally smooth if all maximal chains in its subgroup lattice L(G) are smooth. In this article, we study the product of finite groups which have a permutable subgroup of prime order under the assumption that the maximal subgroups are totally smooth.

Some Results on Products of Finite Groups

Subgroups \(A\) and \(B\) of a finite group are said to be mutually permutable (respectively, M-permutable and \({{\mathrm{sn}}}\)-permutable) if \(A\) permutes with every subgroup (respectively, every maximal subgroup and every subnormal subgroup) of \(B\) and viceversa. If every subgroup of \(A\) permutes with every subgroup of \(B\), then the product is said to be totally permutable. These kinds of products have received much attention in the last twenty years. The aim of this paper is to analyse the behaviour of finite pairwise mutually permutable, mutually M-permutable, mutually \({{\mathrm{sn}}}\)-permutable and totally permutable products with respect to certain classes of groups including the supersoluble groups, widely supersoluble groups, and also the classes of \(PST\)-, \(PT\)- and \(T\)-groups.

Fusion Procedure for Wreath Products of Finite Groups by the Symmetric Group

Let G be a finite group. A complete system of pairwise orthogonal idempotents is constructed for the wreath product of G by the symmetric group by means of a fusion procedure, that is by consecutive evaluations of a rational function with values in the group ring. This complete system of idempotents is indexed by standard Young multi-tableaux. Associated to the wreath product of G by the symmetric group, a Baxterized form for the Artin generators of the symmetric group is defined and appears in the rational function used in the fusion procedure.

On mutually permutable products of finite groups

We say that the subgroups G1 and G2 of a group G are mutually permutable if G1 permutes with every subgroup of G2 and G2 permutes with every subgroup of G1. Let G=G1G2…Gn be the product of its pairwise permutable subgroups G1,G2,…,Gn such that the product GiGj is mutually permutable. We investigate the structure of the finite group G if special properties of the factors G1,G2,…,Gn are known. Our results improve and extend some results of Asaad and Shaalan [1], Ezquerro and Soler-Escriva [9] and Asaad and Monakhov [3].

Indecomposable Projective Representations of Direct Products of Finite Groups Over a Field of Characteristic p

Let K be a field of characteristic p > 0, K* the multiplicative group of K and G = G p × B a finite group, where G p is a p-group and B is a p′-group. Denote by K λ G a twisted group algebra of G over K with a 2-cocycle λ ∈Z 2(G, K*). In this article, we give necessary and sufficient conditions for K λ G to be of OTP representation type, in the sense that every indecomposable K λ G-module is isomorphic to the outer tensor product V#W of an indecomposable K λ G p -module V and an irreducible K λ B-module W.

Mutually Permutable Products of Finite Groups

Let G be a finite group and G1, G2 are two subgroups of G. We say that G1 and G2 are mutually permutable if G1 is permutable with every subgroup of G2 and G2 is permutable with every subgroup of G1. We prove that if is the product of three supersolvable subgroups G1, G2, and G3, where Gi and Gj are mutually permutable for all i and j with and the Sylow subgroups of G are abelian, then G is supersolvable. As a corollary of this result, we also prove that if G possesses three supersolvable subgroups whose indices are pairwise relatively prime, and Gi and Gj are mutually permutable for all i and j with , then G is supersolvable.

On finite products of groups and supersolubility

Two subgroups X and Y of a group G are said to be conditionally permutable in G if X permutes with Y g for some element g ∈ G , i.e., X Y g is a subgroup of G . Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.

Some criteria for supersolubility in products of finite groups

Let H and T be subgroups of a finite group G. H is said to be permutable with T in G if HT = TH. In this paper, we use the concept of permutable subgroups to give two new criterions of supersolubility of the product G = AB of finite supersoluble groups A and B.

On rationality and 2-reflexiveness of wreath products of finite groups

A finite group G is said to be rational if each its irreducible character acquires only rational values, and it is said to be 2-reflexive if each its element can be represented as a product of at most two involutions. We find necessary and sufficient conditions for the wreath of two finite groups be rational and 2-reflexive. Namely, we show that the wreath H ≀ K of two finite groups H and K is a rational (respectively, 2-reflexive) group iff H is a rational (respectively, 2-reflexive) group and K is an elementary Abelian 2-group. As a corollary, we obtain a description of all classical linear groups over finite fields of odd characteristic with rational and 2-reflexive Sylow 2-subgroups.

Residual properties of free products of finite groups

Using a probabilistic approach we establish a new residual property of free products of finite groups.