Related papers: Counting subgroups of fixed order in finite abelia…

The main goal of this paper is to apply the arithmetic method developed in our previous paper \cite{13} to determine the number of some types of subgroups of finite abelian groups.

We prove by using simple number-theoretic arguments formulae concerning the number of elements of a fixed order and the number of cyclic subgroups of a direct product of several finite cyclic groups. We point out that certain multiplicative…

This paper deals with the number of subgroups of a given exponent in a finite abelian group. Explicit formulas are obtained in the case of rank two and rank three abelian groups. An asymptotic formula is also presented.

In this short note we give a formula for the number of chains of subgroups of a finite elementary abelian $p$-group. This completes our previous work [5].

Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, $p$-radical subgroups typically play a critical role. To classify the…

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…

The article presents several methods for the arithmetic of finite abelian groups. We introduce a tool - already used by Delsarte in [1] as I found out later - analogous to Dirichlet's convolution to obtain combinatorial results on these…

We consider abelain subgroups of small index in finite groups. More generally, we consider subgroups such that the product of their index by the index of their centralizer is small.

In this short note we give a formula for the factorization number $F_2(G)$ of a finite rank $3$ abelian $p$-group $G$. This extends a result in our previous work \cite{9}.

We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields…

We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral…

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

Let KG be a group algebra of a finite p-group G over a finite field K of characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial 2-group or the central product of such a group…

A notion of degeneration of elements in groups is introduced. It is used to parametrize the orbits in a finite abelian group under its full automorphism group by a finite distributive lattice. A pictorial description of this lattice leads…

We classify abelian subgroups of Out(F_n) up to finite index in an algorithmic and computationally friendly way. A process called disintegration is used to canonically decompose a single rotationless element \phi into a composition of…

Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) \leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing…

Suppose that $G$ is a finite $p$-group. If all subgroups of index $p^t$ of $G$ are abelian and at least one subgroup of index $p^{t-1}$ of $G$ is not abelian, then $G$ is called an $\mathcal{A}_t$-group. In this paper, some information…

For a finite group $G$, we associate the quantity $\beta(G)=\frac{|L(G)|}{|G|}$, where $L(G)$ is the subgroup lattice of $G$. Different properties and problems related to this ratio are studied throughout the paper. We determine the second…

We formulate the Root Extraction problem in finite Abelian $p$-groups and then extend it to generic finite Abelian groups. We provide algorithms to solve them. We also give the bounds on the number of group operations required for these…

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian $p$-groups with an abelian subgroup of index $p$. In particular, this gives many new examples illustrating the enormous variety…