International Journal of Group Theory
Volume:1 Issue: 3, Sep 2012

Let $G$ be a finite group and let $text{cd}(G)$ be the set of all complex irreducible character degrees of $G$. B. Huppert conjectured that if $H$ is a finite nonabelian simple group such that $text{cd}(G) =text{cd}(H)$, then $Gcong H times A$, where $A$ is an abelian group. In this paper, we verify the conjecture for $rm{F_4(2)}.$

Let $G$ be a group and $A = Aut(G)$ be the group of automorphisms of $G$. Then the element $[g,alpha] = g^{-1}alpha(g)$ is an au- tocommutator of $gin G$ and $alphain A$. Also, the autocommutator subgroup of G is defined to be $K(G) =< [g,alpha] gin G, alphain A >$, which is a characteristic subgroup of G containing the derived sub- group $G'$ of $G$. A group is defined as A-perfect, if it equals its own autocommutator subgroup. The present research is aimed at classifying finite abelian groups which are A-perfect.

Let G be a finite group and pi_{e}(G) be the set of element orders of G. Let k in pi_{e}(G) and m_{k} be the number of elements of order k in G. Let nse(G):={m_{k} | k in pi_{e}(G)}. In this paper, we prove that if G is a group such that nse(G)=nse(PSL(2, 25)), then G is isomorphic to PSL(2, 25).

In this paper, we investigate the influence of some subgrops of Sylow subgroups with semi cover-avoiding property and F-supplementation on the structure of finite groups and generalize a series of known results.

Recently, the authors gave some conditions under which a direct product of finitely many groups is V-capable if and only if each of its factors is V-capable for some varieties V. In this paper, we extend this fact to any infi nite direct product of groups. Moreover, we conclude some results for V-capability of direct products of infi nitely many groups in varieties of abelian, nilpotent and polynilpotent group

A subgroup $X$ of a group $G$ is almost normal if the index $|G:N_G(X)|$ is finite, while $X$ is nearly normal if it has finite index in the normal closure $X^G$. This paper investigates the structure of groups in which every (infinite) subgroup is either almost normal or nearly normal.

Let G be a finite group and let Gamma(G) be the prime graphof G. Assume 2 < q = p^{alpha} < 100. We determine finite groupsG such that Gamma(G) = Gamma(U_3(q)) and prove that if q neq3, 5, 9, 17, then U_3(q) is quasirecognizable by prime graph,i.e., if G is a finite group with the same prime graph as thefinite simple group U_3(q), then G has a unique non-Abeliancomposition factor isomorphic to U_3(q). As a consequence of ourresults, we prove that the simple groups U_{3}(8) and U_{3}(11)are 4-recognizable and 2-recognizable by prime graph,respectively. In fact, the group U_{3}(8) is the first examplewhich is a 4-recognizable by prime graph.