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 $nu(G)=3$.

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 $A(4)cong 2^7{:}Sp_6(2)$ 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 $gamma(S_n)$ be the minimum number of proper subgroups $H_i, i=1, dots, l $ of the symmetric group $S_n$ such that each element in $S_n$ lies in some conjugate of one of the $H_i.$ In this paper we conjecture that $$gamma(S_n)=frac{n}{2}left(1-frac{1}{p_1}right) left(1-frac{1}{p_2}right)+2,$$ where $p_1,p_2$ are the two smallest primes in the factorization of $ninmathbb{N}$ and $n$ is neither a prime power nor a product of two primes. Support for the conjecture is given by a previous result for $n=p_1^{alpha_1}p_2^{alpha_2},$ with $(alpha_1,alpha_2)neq (1,1)$. We give further evidence by confirming the conjecture for 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 $gamma(A_n)$, when $n$ is even, and provide a similar amount of evidence.