INTERNATIONAL JOURNAL OF GROUP THEORY
Year:2013 | Volume:2 | Issue:2|Papers Count:7

The number of factorizations of a finite abelian group as the product of two subgroups is computed in two different ways and a combinatorial identity involving Gaussian binomial coefficients is presented.

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 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 present some results about subgroup which is generalization of the sub- group R2Ä(G)={aÎG|[a,g]Ä g=1Ä, "gÎG} of right 2Ä-Engel elements of a given group G. If p is an odd prime, then with the help of these results, we obtain some results about tensor squares of p-groups satisfying the law [x,g,y]Äg=1Ä, for all x,g,yÎG. In particular p-groups satisfying the law [x,g,y]Äg=1Ä have abelian tensor squares. Moreover, we can determine tensor squares of two-generator p-groups of class three satisfying the law [x,g,y]Äg=1Ä.

Let G be a finite group. We denote by ψ(G) the integer ågÎGo(g), where o(g) denotes the order of gÎG. Here we show that ψ(A5)<ψ(G) for every non-simple group G of order 60, where A5 is the alternating group of degree 5. Also we prove that ψ(PSL (2,7))<ψ(G) for all non-simple groups G of order 168. These two results confirm the conjecture posed in [J. Algebra Appl., 10 No.2 (2011) 187-190] for simple groups A5 and PSL (2,7).

Abstract. Three in finite families of finite abelian groups will be described such that each member of these families has the Redei k-property for many non-trivial values of k.

The non-commuting graph Ñ(G) of a non-abelian group G is defined as follows: its vertex set is G-Z (G) and two distinct vertices x and y are joined by an edge if and only if the commutator of x and y is not the identity. In this paper we prove that if G is a finite group with Ñ(G)@Ñ(Sn), then G@Sn, where Sn is the symmetric group of degree n, where n is a natural number.