Related papers: Generating alternating and symmetric groups with t…

Let B(n) be the set of pairs of permutations from the symmetric group of degree n with a 3-cycle commutator, and let A(n) be the set of those pairs which generate the symmetric or the alternating group of degree n. We find effective…

A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response…

Let $n$ be a positive integer, $\sigma$ be an element of the symmetric group $\mathcal{S}_n$ and let $\sigma$ be a cycle of length $n$. The elements $\alpha ,\beta \in \mathcal{S}_n$ are $\sigma$-equivalent, if there are natural numbers $k$…

In the note we prove that all composition factors of a finite group possessing a Carter subgroup of odd order either are abelain, or are isomorphic to $L_2(3^{2n+1})$.

In this paper, we introduce a novel approach for generating random elements of a finite group given a set of generators of that. Our method draws upon combinatorial group theory and automata theory to achieve this objective. Furthermore, we…

We prove that every finite simple group of Lie type $G$ can be generated by three regular unipotent elements. In certain cases we show that two regular unipotents are sufficient to generate $G$.

Two finite words $u$ and $v$ are $k$-binomially equivalent if, for each word $x$ of length at most $k$, $x$ appears the same number of times as a subsequence (i.e., as a scattered subword) of both $u$ and $v$. This notion generalizes…

This paper examines order three elements of finite groups which normalize no nontrivial 2-subgroup. The motivation for finding such elements arises out of a problem in modular representation theory. The question of when these elements…

Let $\mathbb{A} = (A, \cdot)$ be a semigroup. We generalize some recent results by G. A. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable…

We extend to alternating groups $A_n$ several results about symmetric groups asserting that under various conditions on a conjugacy class, or more generally, a normal subset, $C$ of $S_n$, we have $C^2 \supseteq A_n\setminus\{1\}$

Given a subset $S$ of the non-identity elements of the dihedral group of order $2m$, is it possible to order the elements of $S$ so that the partial products are distinct? This is equivalent to the sequenceability of the group when $|S| =…

We prove a conjecture dating back to a 1978 paper of D.R.\ Musser~\cite{musserirred}, namely that four random permutations in the symmetric group $\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > \epsilon$ for some…

Two finite groups are said to have the same order type if for each positive integer $n$ both groups have the same number of elements of order $n$. In 1987 John G. Thompson asked if in this case the solvability of one group implies the…

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form $n$ over a field $k$ of characteristic not two, and a category arising from an action…

Description of adjoint invariants of general Linear Lie superalgebras $\mathfrak{gl}(m|n)$ by Kantor and Trishin is given in terms of supersymmetric polynomials. Later, generators of invariants of the adjoint action of the general linear…

We give an explicit geometric argument that Artin's braid group $B_n$ is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical…

We prove several new results on the structure of the subgroup generated by a small doubling subset of an ordered group, abelian or not. We obtain precise results generalizing Freiman's 3k-3 and 3k-2 theorems in the integers and several…

Let $G$ be a finite group and $N$ a proper, nontrivial, normal subgroup of $G$. If, for every element $x$ of $G$ not lying in $N$, the elements in the coset $xN$ all have the same order as $x$, then we say that $(G,N)$ is an {\it{equal…

This paper discusses the permutations that are generated by rotating $k \times k$ blocks of squares in a union of overlapping $k \times (k+1)$ rectangles. It is found that the single-rotation parity constraints effectively determine the…

All possible products of all elements of an odd order finite group are considered. A set of all such products is called as a K-set. A hypothesis of K-set coincidence of any group of an odd order with its commutant is proposed and the…