Let G be a finite group. We say that the group G has the Camina triple property if there exist normal subgroups M and N of G with N M such that for every g G M  \ the conjugacy class of g in G is equal to gN. In this case,  G M N, ,  is called the Camina triple. In the special case when M N G    , the group G is a Camina group. This shows that the Camina triple property is a generalization of the Camina property in finite groups. In recent years, the groups with the Camina triple property have been studied in some special cases...

Let G be a finite group and let N be a normal subgroup of G. Suppose that Irr(G|N) is the set of the IRREDUCIBLE CHARACTERS of G that contain N in their kernels. In this paper, we classify solvable groups G in which the set C(G)={Irr(G|N)|¹N⊴G} has at most three elements. We also compute the set C(G) for such groups.

In this paper we consider finite groups G satisfying the following condition: G has two columns in its character table which differ by exactly one entry. It turns out that such groups exist and they are exactly the finite groups with a non-trivial intersection of the kernels of all but one IRREDUCIBLE CHARACTERS or, equivalently, finite groups with an IRREDUCIBLE character vanishing on all but two conjugacy classes. We investigate such groups and in particular we characterize their subclass, which properly contains all finite groups with non-linear CHARACTERS of distinct degrees, which were characterized by Berkovich, Chillag and Herzog in 1992.

For a character ,of a , nite group G, the number , c(1) = [G: ker, ] , (1) is called the co-degree of , . Let Sol(G) denote the solvable radical of G. In this paper, we show that if G is a , nite non-solvable group with f, c(1)2: ,2 Irr(G)g = f1,2mg for some positive integer m, then G=Sol(G) has a normal subgroup M=Sol(G) such that M=Sol(G) , = PSL2(2n) for some integer n ,2, [G: M] is odd and G=Sol(G). Aut(PSL2(2n)).

The aim of this paper is to classify the finite simple groups with the number of zeros at most seven greater than the number of nonlinear IRREDUCIBLE CHARACTERS in the character tables. We find that they are exactly A5 , L2 (7) and A6.

Let Irr(G) be the set of IRREDUCIBLE complex CHARACTERS of a finite group G and cd(G)={x(1) | x Î 2 Irr(G)} be the set of IRREDUCIBLE character degrees of G. The question: Given the set cd(G), what can be said about the structure of G? have been studied by several people. In case cd(G)={1,m}, for some integer m, the basic tools for studying the question can be found in Chapter 12 of the well known book of Isaacs [8]. Isaacs proved in Theorem 12.5 and Corollary 12.6 the following results. Theorem A. Let G be a finite group and cd(G) = {1,m}. Then, at least one of the followings occurs: (a) G has an abelian normal subgroup of index m.

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.

Let R be a commutative ring with identity and M be an R-module. In this paper, we will introduce the concept of 2-IRREDUCIBLE (resp., strongly 2-IRREDUCIBLE) submodules of M as a generalization of IRREDUCIBLE (resp., strongly IRREDUCIBLE) submodules of M and investigated some properties of these classes of modules.