Related papers: On reconstructing reducible n-ary quasigroups and …

A group is said to be cube-free if its order is not divisible by the cube of any prime. Let $f_{cf,sol}(n)$ denote the isomorphism classes of solvable cube-free groups of order $n$. We find asymptotic bounds for $f_{cf,sol}(n)$ in this…

A conjecture of Rosenberger says that a group of the form $\langle x,y|x^p=y^q=W(x,y)^r=1\rangle$ (with $r>1$) is either virtually solvable or contains a non-abelian free subgroup. This note is an account of an attack on the conjecture in…

A square matrix of order n with $n\geq 2$ is called permutative matrix when all its rows (up to the frst one) are permutations of precisely its frst row. In this paper recalling spectral results for partitioned into $2$-by-$2$ symmetric…

Given a quasiperiodic cocycle in sl(2, R) sufficiently close to a constant, we prove that it is almost-reducible in ultradifferentiable class under an adapted arithmetic condition on the frequency vector. We also give a corollary on the…

An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element $s$ in a dual group such that $s^2=1$. We prove that there is a bijection…

We prove that any quasigroup admissing complete or quasicomplete mapping has a prolongation to a quasigroup having one element more.

For a finite abelian group $G$ and a positive integer $k$, let $s_{k}(G)$ denote the smallest integer $\ell\in\mathbb{N}$ such that any sequence $S$ of elements of $G$ of length $|S|\geq\ell$ has a zero-sum subsequence with length $k$. The…

Let $N(d,n)$ be the variety of all $d$-tuples of commuting nilpotent $n\times n$ matrices. It is well-known that $N(d,n)$ is irreducible if $d=2$, if $n\le 3$ or if $d=3$ and $n=4$. On the other hand $N(3,n)$ is known to be reducible for…

Let $K=Z/pZ$ and let $A$ be a subset of $\GL_r(K)$ such that $<A>$ is solvable. We reduce the study of the growth of $A$ under the group operation to the nilpotent setting. Specifically we prove that either $A$ grows rapidly (meaning…

Let $(G,+)$ be an abelian group and consider a subset $A \subseteq G$ with $|A|=k$. Given an ordering $(a_1, \ldots, a_k)$ of the elements of $A$, define its {\em partial sums} by $s_0 = 0$ and $s_j = \sum_{i=1}^j a_i$ for $1 \leq j \leq…

This is an expository work presenting in detail the proof of the structure theorem for divisible abelian groups. A divisible abelian group is an abelian group that satisfies nD=D for all natural n. The theorem states that any divisible…

Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2'$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive…

We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has…

We call an affine algebraic supergroup quasireductive if its underlying algebraic group is reductive. We obtain some results about the structure and representations of reductive supergroups.

We prove that irreducible complex representations of finitely generated nilpotent groups are monomial if and only if they have finite weight, which was conjectured by Parshin. Note that we consider (possibly, infinite-dimensional)…

A group of order $p^n$ ($p$ prime) has an indecomposable polynomial invariant of degree at least $p^{n-1}$ if and only if the group has a cyclic subgroup of index at most $p$ or it is isomorphic to one of two particular groups of small…

Let $H$ be a finite quasisimple classical group, i.e. $H$ is perfect and $S:=H/Z(H)$ is a finite simple classical group. We prove in this paper that, excluding the cases when the simple group $S$ has a very exceptional Schur multiplier such…

There are several results in the literature concerning $p$-groups $G$ with a maximal elementary abelian normal subgroup of rank $k$ due to Thompson, Mann and others. Following an idea of Sambale we obtain bounds for the number of generators…

An ordered triple $(s,p,n)$ is called admissible if there exist two different multisets $X=\{x_1,x_2,\dotsc,x_n\}$ and $Y=\{y_1,y_2,\dotsc,y_n\}$ such that $X$ and $Y$ share the same sum $s$, the same product $p$, and the same size $n$. We…

A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups…