We establish correspondances between factorisations of finite abelian groups ( direct factors, unitary factors, non isomorphic subgroup classes ) and factorisations of integer matrices. We then study counting functions associated to these factorisations and find average orders. Mathematics Subject Classification 11M41,20K01,15A36.

One of the main directions in group theory is study impact characteristic subgroups on structure whole group. Such include different $\Sigma$-norms a A $\Sigma$-norm intersection normalizers all system $\Sigma$. The authors non-periodic groups with restrictions such -- norm $N_{G}(C_{\bar{p}})$ cyclic non-prime order, which composite or infinite order $G$. It was proved that if $G$ mixed group,...

A residually finite (profinite) group G is just infinite if every non-trivial (closed) normal subgroup of G is of finite index. This paper considers the problem of determining whether a (closed) subgroup H of a just infinite group is itself just infinite. If G is not virtually abelian, we give a description of the just infinite property for normal subgroups in terms of maximal subgroups. In par...

‎In this paper we prove that a finite group $G$ having at most three‎ ‎conjugacy classes of non-normal non-abelian proper subgroups is‎ ‎always solvable except for $Gcong{rm{A_5}}$‎, ‎which extends Theorem 3.3‎ ‎in [Some sufficient conditions on the number of‎ ‎non-abelian subgroups of a finite group to be solvable‎, ‎Acta Math‎. ‎Sinica (English Series) 27 (2011) 891--896.]‎. ‎Moreover‎, ‎we s...

Let $G$ be a $p$-group of order $p^n$ and $Phi$=$Phi(G)$ be the Frattini subgroup of $G$. It is shown that the nilpotency class of $Autf(G)$, the group of all automorphisms of $G$ centralizing $G/ Fr(G)$, takes the maximum value $n-2$ if and only if $G$ is of maximal class. We also determine the nilpotency class of $Autf(G)$ when $G$ is a finite abelian $p$-group.

We consider the following conjecture. Suppose that G is a non-free non-cyclic onerelator group such that each subgroup of finite index is again a one-relator group and each subgroup of infinite index is a free group. Must G be a surface group? We show that if G is a freely indecomposable fully residually free group and satisfies the property that every subgroup of infinite index is free then G ...

‎let $v$ be a vector space over a field $f$ of characteristic $pgeq 0$ and let‎ ‎$t$ be a regular subgroup of the affine group $agl(v)$‎. ‎in the finite dimensional case we show that‎, ‎if $t$ is abelian or $p>0$‎, ‎then $t$ is unipotent‎. ‎for $t$ abelian‎, ‎pushing forward some ideas used in [a‎. ‎caranti‎, ‎f‎. ‎dalla volta and m‎. ‎sala‎, ‎abelian regular subgroups of the affine group and r...