In their paper "Finite groups whose noncentral commuting elements have centralizers of equal size", S. Dolfi, M. Herzog and E. Jabara classify the groups in question- which they call CH-groups- up to finite p-groups. Our goal is to investigate the finite p-groups in the class. The chief result is that a finite p-group that is a CH -group either has an abelian maximal subgroup or is of class at most p+1. Detailed descriptions, in some cases characterisations up to isoclinism, are given.

Introduction: Let G be a finite group and A be a normal subgroup of G. We denote by nee(A) the number of G-conjugacy classes of A and A is called n-decomposable, if ncc (A) = n. Set Ncc(G) = {ncc(A) | A D G}. Let X be a nonempty subset of positive integers. A group G is called X-decomposable, if Nee(G) =X.Aim: The aim of this paper is finding all subsets X of positive integers such that there is a X-decomposable finite group.Materials and Methods: In this paper, by using a method of Shahryari and Shahabi given in [10], we first obtain the center of the groups under consideration and then by applying some results of Karpilovsky, the characterization of the groups is obtained.Results: We first characterize the solvable n-decomposable finite groups and then by using some results of Breuer in the classification of finite simple groups, the characterization of n-decomposable finite groups, 1£n£8, are given.Conclusion: Using the methods presented here and some complicated calculations, it is possible to solve the problem for n£12.

Let F be a finite extension of Q, Qp or a global field of positive characteristic, and let E/F be a Galois extension. We study the realization fields of finite subgroups G of GLn(E) stable under the natural operation of the Galois group of E/F. Though for sufficiently large n and a fixed algebraic number field F every its nite extension E is realizable via adjoining to F the entries of all matrices gÎG for some finite Galois stable subgroup G of GLn(C), there is only a finite number of possible realization field extensions of F if GÌGLn(OE) over the ring OE of integers of E. After an exposition of earlier results we give their refinements for the realization fields E/F. We consider some applications to quadratic lattices, arithmetic algebraic geometry and Galois cohomology of related arithmetic groups.

Let G be a finite group. A subset X of G is a set of pairwise non-commuting elements if any two distinct elements of X do not commute. In this paper we determine the maximum size of these subsets in any finite non-abelian metacyclic 2-group and in any finite non-abelian p-group with an abelian maximal subgroup.

Let G be a finite group and let P=P1, …; Pm be a sequence of Sylow pi -subgroups of G, where p1;…; pm are the distinct prime divisors of |G|. The Sylow multiplicity of g 2 G in P is the number of distinct factorizations g=g1 … gm such that gi Î Pi. We review properties of the solvable radical and the solvable residual of G which are formulated in terms of Sylow multiplicities, and discuss some related open questions.

A group G is said to be a C-tidy group if for every element x Î G \ K (G), the set Cyc (x) ={y Î G | áx, y ñ is cyclic} is a cyclic subgroup of G, where K (G) = ÇxÎG Cyc (x). In this short note we determine the structure of finite C-tidy groups.

We characterize the rational subsets of a finite group and discuss the relations to integral Cayley graphs.

We say that a finite group G is conjugacy expansive if for any normal subset S and any conjugacy class C of G the normal set SC consists of at least as many conjugacy classes of G as S does. Halasi, Maroti, Sidki, Bezerra have shown that a group is conjugacy expansive if and only if it is a direct product of conjugacy expansive simple or abelian groups.By considering a character analogue of the above, we say that a finite group G is character expansive if for any complex character and irreducible character c of G the character ac has at least as many irreducible constituents, counting without multiplicity, as a does. In this paper we take some initial steps in determining character expansive groups.

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For a  nite group G and a positive integer n, let G(n) be the set of all elements in G such that xn = 1. The groups G and H are said to be of the same (order) type if jG(n)j = jH(n)j, for all n. The main aim of this paper is to show that if G is a  nite group of the same type as Suzuki groups Sz(q), where q = 22m+1  8, then G is isomorphic to Sz(q). This addresses to the well-known J. G. Thompson's problem (1987) for simple groups.