International Journal of Group Theory
Volume:3 Issue: 2, June 2014

For a finite group $G$ let $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$. The aim of this paper is to classify all the non-nilpotent groups with $nu(G)=3$.

We exhibit an explicit construction for the second cohomology group $H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$. We show how the elements of $H^2(L, A)$ correspond one-to-one to the equivalence classes of central extensions of $L$ by $A$, where $A$ now is considered as an abelian Lie ring. For a finite Lie ring $L$ we also show that $H^2(L, C^*) cong M(L)$, where $M(L)$ denotes the Schur multiplier of $L$. These results match precisely the analogue situation in group theory.

The split extension group A(4) = 2^7:Sp_6(2) is the affine subgroup of the symplectic group Sp_8(2) of index 255. In this paper, we use the technique of the Fischer-Clifford matrices to construct the character table of the inertia group 2^7:(2^5:S_6) of A(4) of index 63.

We characterize those groups G and vector spaces V such that V is a faithful irreducible G-module and such that each v in V is centralized by a G-conjugate of a fixed non-identity element of the Fitting subgroup F(G) of G. We also determine those V and G for which V is a faithful quasi-primitive G-module and F(G) has no regular orbit. We do use these to show in some cases that a non-vanishing element lies in F(G).

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

Let gamma(Sn) be the minimum number of proper subgroups Hi, i = 1,...,ell, of the symmetric group Sn such that each element in Sn lies in some conjugate of one of the Hi. In this paper we conjecture that gamma(Sn) =(n/2)(1-1/p_1) (1-1/p_2) + 2, where p1, p2 are the two smallest primes in the factorization of n and n is neither a prime power nor a product of two primes. Support for the conjecture is given by a previous result for the case where n has at most two distinct prime divisors. We give further evidence by confirming the conjecture for certain integers of the form n = 15q, for an infinite set of primes q, and by reporting on a Magma computation. We make a similar conjecture for gamma(An), when n is even, and provide a similar amount of evidence.