Related papers: Primitive groups, road closures, and idempotent ge…

A finite group $G$ is \emph{coprimely-invariably generated} if there exists a set of generators $\{g_1, ..., g_u\}$ of $G$ with the property that the orders $|g_1|, ..., |g_u|$ are pairwise coprime and that for all $x_1, ..., x_u \in G$ the…

A celebrated result of J. Thompson says that if a finite group $G$ has a fixed-point-free automorphism of prime order, then $G$ is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier…

Let $G$ be a finite permutation group on $\Omega$. An ordered sequence of elements of $\Omega$, $(\omega_1,\dots, \omega_t)$, is an irredundant base for $G$ if the pointwise stabilizer $G_{(\omega_1,\dots, \omega_t)}$ is trivial and no…

Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If…

Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group…

Let $\mathcal{D} = (\mathcal{P}, \mathcal{B})$ be a $2$-$(v, k, \lambda)$ design, and let $G$ be a half-flag-transitive automorphism group of ${\cal D}$. In this article, we first establish three sufficient conditions for $G$ to be…

Let $G$ be a finite group. For all $a \in \Z$, such that $(a,|G|)=1$, the function $\rho_a: G \to G$ sending $g$ to $g^a$ defines a permutation of the elements of $G$. Motivated by a recent generalization of Zolotarev's proof of classic…

We show that the semidirect product of a group $C$ by $A*_D B$ is isomorphic to the free product of $A\rtimes C$ and $B\rtimes C$ amalgamated at $D\rtimes C$, where $A$, $B$ and $C$ are arbitrary groups. Moreover, we apply this theorem to…

The second author introduced notions of weak permutability and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite groups. Two groups H,K weakly commute provided there exists a…

Let $A$ be a finite dimensional commutative associative algebra with unit over an algebraically closed field of characteristic zero. The group $G(A)$ of invertible elements is open in $A$ and thus $A$ has a structure of a prehomogeneous…

Let $R$ be a finite commutative ring with unity $1_R$ and $k \in R$. Properties of one-sided $k$-orthogonal $n \times n$ matrices over $R$ are presented. When $k$ is idempotent, these matrices form a semigroup structure. Consequently new…

Let $G$ be a finite group. Let $K/k$ be a Galois extension of number fields with Galois group isomorphic to $G$, and let $C \subseteq \mathrm{Gal}(K/k) \simeq G$ be a conjugacy invariant subset. It is well known that there exists an…

Let $S$ be a semigroup (written multiplicatively). Endowed with the operation of setwise multiplication induced by $S$ on its parts, the non-empty subsets of $S$ form themselves a semigroup, denoted by $\mathcal P(S)$. Accordingly, we say…

Let $F$ be a field with at least three elements and $G$ a locally finite group. This paper aims to show that if either $F$ is algebraically closed or the characteristic of $F$ is positive, then an element in the group algebra $FG$ is a…

An approach to representations of finite groups is presented without recourse to character theory. Considering the group algebra C[G] as an algebra of linear maps on C[G] (by left multiplication), we derive the primitive central idempotents…

This paper is about the structure of infinite primitive permutation groups and totally disconnected locally compact groups ("tdlc groups'"). The permutation groups we investigate are subdegree-finite (i.e. all orbits of point stabilisers…

Commutative semirings with divisible additive semigroup are studied. We show that an additively divisible commutative semiring is idempotent, provided that it is finitely generated and torsion. In case that a one-generated additively…

A nonpolycyclic nilpotent-by-cyclic group Gamma can be expressed as the HNN extension of a finitely-generated nilpotent group N. The first main result is that quasi-isometric nilpotent-by-cyclic groups are HNN extensions of quasi-isometric…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, assume that $G$ has a maximal $A$-invariant subgroup $M$ that is a direct product of some isomorphic simple groups, we prove that if $G$ has a…

A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_1$ and $L_2$ with $L_1$ not semiprimitive, we construct an infinite…