Related papers: Characterizing finite $p$-groups by their Schur mu…

Let $H\leq K$ be subgroups of a group G. We say that H is strongly closed in K with respect to G if whenever $a^g \in K$ where $a \in H, g \in G,$ then $a^g \in H.$ In this paper, we investigate the structure of a group G under the…

Given a group $G$ and an integer $n\geq2$ we construct a new group $\tilde{{\cal K}}(G,n)$. Although this construction naturally occurs in the context of finding new invariants for complex algebraic surfaces, it is related to the theory of…

We establish conditions under which the fundamental group of a graph of finite $p$-groups is necessarily residually $p$-finite. The technique of proof is independent of previously established results of this type, and the result is also…

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

H\'ethelyi and K\"ulshammer showed that the number of conjugacy classes $k(G)$ of any solvable finite group $G$ whose order is divisible by the square of a prime $p$ is at least $(49p+1)/60$. Here an asymptotic generalization of this result…

We introduce a general constructive method to find a p-basis (and the Ulm invariants) of a finite Abelian p-group M. This algorithm is based on Groebner bases theory. We apply this method to determine the additive structure of…

Necessary and sufficient conditions are given for a $G$-graded simple module over a unital associative algebra, graded by an abelian group $G$, to be isomorphic to a loop module of a simple module, as well as for two such loop modules to be…

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal…

R. Baer has proved that if the factor-group G/{\zeta}_{n}(G) of a group G by the member {\zeta}_{n}(G) of its upper central series is finite (here n is a positive integer) then the member {\gamma}_{n+1}(G) of the lower central series of G…

It has been proved recently by Moreto and Craven that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups…

We give alternative computations of the Schur multiplier of $Sp(2g,\mathbb Z/D\mathbb Z)$, when $D$ is divisible by 4 and $g\geq 4$: a first one using $K$-theory arguments based on the work of Barge and Lannes and a second one based on the…

For a positive integer r we prove that if G is a profinite group in which the centralizer of every nontrivial element has rank at most r, then G is either a pro-p group or a group of finite rank. Further, if G is not virtually a pro-p…

In 1956, Green provided a bound on the order of the Schur multiplier of $p$-groups. This bound, given as a function of the order of the group, is the best possible. Since then, the bound has been refined numerous times by adding other…

A subset P of N x N is called Schur bounded if every infinite matrix with bounded entries which is zero off of P yields a bounded Schur multiplier on B(H). Such sets are characterized as being the union of a subset with at most k entries in…

A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if G is an abelian group, then the follwing are equivalent: 1. Th(G, +) has the…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

For $G$ an algebraic group of type $A_l$ over an algebraically closed field of characteristic $p$, we determine all irreducible rational representations of $G$ in defining characteristic with dimensions $\le (l+1)^s$ for $s = 3, 4$,…

We prove that if $G$ is a totally bounded abelian group \st\ its dual group $\widehat{G}_p$ equipped with the finite-open topology is a Baire group, then every compact subset of $G$ must be finite. This solves an open question by Chasco,…

A sequence $(a_{n}) $ in an Abelian group is called a $T$-sequence if there exists a Hausdorff group topology on $G$ in which $(a_{n}) $ converges to $0$. For a $T$-sequence $(a_{n}) $, $\tau_{(a_{n}) } $ denotes the strongest group…

This article determines the structure of the group ring $\mathbb{Z}_nG$, where $G$ is a finite group and $\mathbb{Z}_n$ is the ring of integers modulo $n$, such that $n$ is relatively prime to the order of $G$. The decomposition of…