On Finite Groups with Special Conjugacy Classes

Let G be a finite group with the property that for any conjugacy class order, G has exactly two conjugacy classes which have the same order. We prove that： （1） ff a Sylow 2-subgroup of G is Abelian, then G is isomorphic to the direct product of symmetric group with order 3 and cyclic group with order 2, or G is isomorphic to the semidirect product of a cyclic group with order 3 and a cyclic group with order 4; （2） if G＇ is nilpotent, then G is a group of {2,3,5 }.

Conjugacy Classes and Characters for Extensions of Finite Groups

Let H be an extension of a finite group Q by a finite group G. Inspired by the results of duality theorems for etale gerbes on orbifolds, the authors describe the number of conjugacy classes of H that map to the same conjugacy class of Q. Furthermore, a generalization of the orthogonality relation between characters of G is proved.

Curve selection Lemma for semianalytic sets and conjugacy classes of finite order in Lie groups

Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group G, each connected component of the set E_n(G)of all elements of order n in G is a conjugacy class in G. In particular, all conjugacy classes of finite order in G are closed. Some properties of connected components of E_n(G) are also given.

Coprime action and arithmetical conditions on invariant conjugacy classes

Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We show that if 4 divides no A-invariant conjugacy class size of G, then G is solvable. We also characterize the A-invariant structure of G under certain arithmetical conditions on the set of A-invariant class sizes of G by means of the fixed point subgroup, some of which imply the solvability of G. Thus, we extend, for coprime action, several results appeared in the literature on class sizes.

An Arithmetical Condition on the Sizes of Conjugacy Classes of a Finite Group

An element x of a finite group G is said to be primary if the order of x is a prime power.We define csp2(G)as follows:if|x^(G)|is a prime power for every primary element x of G,where x^(G) is the conjugacy class of x in G,then csp2(G)=0;if there exists a primary element x in G such that|x^(G)|is divisible by at least two distinct primes,then csp2(G)=max{|x^(G)||x∈Gis primary,|x^(G)|is divisible by at least two distinct primes}.In this paper we discuss the influence of the number csp2(G)on the structure of G.

COSET DIAGRAMS FOR A HOMOMORPHIC IMAGE OFΔ(3,3,k)

Let q be a prime power. By PL（Fq） the authors mean a projective line over the finite field Fq with the additional point ∞. In this article, the authors parametrize the conjugacy classes of nondegenerate homomorphisms which represent actions of △（3, 3, k） = （u, v： u^3 = v^3 = （uv）^k = 1〉on PL（Fq）, where q ≡ ±1（modk）. Also, for various values of k, they find the conditions for the existence of coset diagrams depicting the permutation actions of △（3, 3, k） on PL（Fq）. The conditions are polynomials with integer coefficients and the diagrams are such that every vertex in them is fixed by （u^-v^-）^k. In this way, they get △（3, 3, k） as permutation groups on PL（Fq）.

A Note on the Prime Spectrums of Character Rings of Finite Groups

Let G be a finite group with order g and S be a subring of the algebraic number field which contains the integral extension over Z generated by a g-th primitive root co of unity, and R（G） be the character ring of G. The prime spectrum of the commutative ring S×Z R（G） iv denoted by Spec（S×Z R（G）） and set π={p｜p is a rational prime number such that p^-1 S}. We prove that when G is a regroup, a π＇-group, or a finite Abelian group, the number of the connetted components of Spec（ S×Z R （G） ） coincides with the number of the π-regular classes in G,

The Class Number of Derived Subgroups and the Structure of Camina Groups

A finite group G is called a Camina group if G has a proper normal subgroup N such that gN is precisely a conjugacy class of G for any g ∈E G - N. In this paper, the structure of a Camina group G is determined when N is a union of 2, 3 or 4 conjugacy classes of G.

On a Graph Related to Subgroups Generated by a Conjugacy Class in Finite Groups

Let G be a group,and let →/ΓC(G) be a digraph whose vertices are the nontrivial conjugacy classes of G and there is an arc from a vertex C to a vertex C' if and only if C≠C' and(C)■(C').In this paper,we characterize finite groups G whose associated digraphs →/ΓC(G) are oriented trees.

On the Normal Subgroup with Exactly Two G-Conjugacy Class Sizes

Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either i or m. Particularly, it is shown that N is Abelian if N ∩ Z（G）=1 and the G-conjugacy class size of every element of N is either 1 or m.

On Conjugacy Class Graph of Normal Subgroup

Let G be a finite group and N a normal subgroup of G.Denote by Γ_(G)(N)the graph whose vertices are all distinct G-conjugacy class sizes of non-central elements in N,and two vertices of Γ_(G)(N)are adjacent if and only if they are not coprime numbers.We prove that if the center Z(N)=Z(G)∩N and Γ_(G)(N)is k-regular for k≥1,then either a section of Nis a quasi-Frobenius group or Γ_(G)(N)is a complete graph with k+1 vertices.

On conjugacy class sizes of primary and biprimary elements of a finite group

Let m, n 〉 1 be two coprime integers. In this paper, we prove that a finite solvable group is nilpotent if the set of the conjugacy class sizes of its primary and biprimary elements is {1, rn, n, mn}.

A Result on Character Degrees and Conjugacy Class Sizes in Finite Groups

Let G be a finite group and π be a set of primes including at least two elements. We write cd（G） and cs（G） to denote the set of complex irreducible character degrees and conjugacy class sizes of G , respectively, and write π（m）to denote the set of all prime divisors of a positive integer m . For any 1≠m∈cd（G） and 1≠m∈cs（G）, in this note, we shall present the corresponding group structures of finite group G in the case π（m）=π , respectively, which generalizes the result of finite groups with character degrees of two distinct primes. Furthermore, we shall see that the influence of the two sets on the group structure is analogous.

Thompson's conjecture for Lie type groups E_7(q)

In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type ET（q）. More precisely, we show that every finite group G with the properties Z（G） = 1 and cs（G） = cs（ET（q）） is necessarily isomorphic to ET（q）, where cs（G） and Z（G） are the set of lengths of conjugacy classes of G and the center of G respectively.

Factorizations of Finite Groups and Related Topics

This is a survey on the recent progress in the theory of finite groups with factorizations and around it,done by the author and his coauthors,and this has no pretensions to cover all topics in this wide area of research.In particular,we only touch the great consequences of the fundamental paper of Liebeck,Praeger and Saxl on maximal factorizations of almost simple finite groups for the theory of groups with factorizations.In each case the reader can find additional references at the end of Section 1.Some of the methods of investigation can be used to obtain information about finite groups in general,nilpotent algebras and related nearrings.

Thompson＇s conjecture for alternating group of degree 22

For a finite group G, it is denoted by N（G） the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture： if L is a finite nonabelian simple group, G is a finite group with trivial center, and N（G） = N（L）, then L and G are isomorphic. In this paper, it is proved that Thompson＇s conjecture is true for the alternating group A22 with connected prime graph.