Related papers: On pyramidal groups whose number of involutions is…

Let $G$ be a group. The prime index graph of $G$, denoted by $\Pi(G)$, is the graph whose vertex set is the set of all subgroups of $G$ and two distinct comparable vertices $H$ and $K$ are adjacent if and only if the index of $H$ in $K$ or…

The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner…

We introduce the notion of a powerfully solvable group. These are powerful groups possessing an abelian series of a special kind. These groups include in particular the class of powerfully nilpotent groups. We will also see that for a…

Let $G$ be a nontrivial permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is…

In [14] Hemmer conjectures that the module of fixed points for the symmetric group $\Sigma_m$ of a Specht module for $\Sigma_n$ (with $n>m$), over a field of positive characteristic $p$, has a Specht series, when viewed as a…

Let $G$ be a group, define an equivalence relation $\sim$ as below: $$\forall \ g, h \in G, \ g \sim h \Longleftrightarrow |g| = |h|$$ the set of sizes of equivalence classes with respect to this relation is called the same-order type of…

We study the class of graphs known as k-trees through the lens of Joyal's theory of combinatorial species (and an equivariant extension known as '$\Gamma$-species' which incorporates data about 'structural' group actions). This culminates…

It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately,…

Let $G$ be a group. An automorphism of $G$ is called intense if it sends each subgroup of $G$ to a conjugate; the collection of such automorphisms is denoted by $\mathrm{Int}(G)$. In the special case in which $p$ is a prime number and $G$…

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…

A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of…

For natural numbers $k<n$ we study the graphs $T_{n,k}:=K_{k}\lor\overline{K_{n-k}}$. For $k=1$, $T_{n,1}$ is the star $S_{n-1}$. For $k>1$ we refer to $T_{n,k}$ as a \emph{graph of pyramids}. We prove that the graphs of pyramids are…

We introduce a special class of powerful $p$-groups that we call powerfully nilpotent groups that are finite $p$-groups that possess a central series of a special kind. To these we can attach the notion of a powerful nilpotence class that…

We obtain a classification of the finite two-generated cyclic-by-abelian groups of prime-power order. For that we associate to each such group $G$ a list $\inv(G)$ of numerical group invariants which determines the isomorphism type of $G$.…

Let $\pi$ be a set of primes. According to H. Wielandt, a subgroup $H$ of a finite group $X$ is called a $\pi$-submaximal subgroup if there is a monomorphism $\phi:X\rightarrow Y$ into a finite group $Y$ such that $X^\phi$ is subnormal in…

In this paper, we construct an infinite families of group $G$ of order $4p^m$ which can be an automorphism group of some regular polytope with type $\{p, 2p\}$, where $m \geq 3$ and $p$ is an odd prime. For $p=3$, our polytopes are the…

If $G$ is a finite group and $k =q>2$ or $k=q+1$ for a prime power $q$ then, for infinitely many integers $v$, there is a $2$-$(v,k,1)$-design ${\bf D}$ for which ${\rm Aut} {\bf D}\cong G$.

We characterise finite groups such that for an odd prime $p$ all the irreducible characters in its principal $p$-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by $p$ unless…

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of…

A group $\Gamma$ is said to be periodic if for any $g$ in $\Gamma$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $\SS^2$ by $C^1$-diffeomorphisms with a finite…