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The symmetry of polygons can be characterized by the number of symmetry axes they have. For $n$-polygons with $p$ or $p^2$ vertices $p\geq3$ there exist few symmetry categories, depending from the number of symmetry-axes the have. Further…

The Grassmannian admits an action by a finite cyclic group via the cyclic shift map. We give a simple description of the points fixed by each element of this cyclic group, extending Karp's description of the points fixed by the cyclic shift…

We consider complete intersection ideals in a polynomial ring over a field of characteristic zero that are stable under the action of the symmetric group permuting the variables. We determine the possible representation types for these…

For almost finite groupoids, we study how their homology groups reflect dynamical properties of their topological full groups. It is shown that two clopen subsets of the unit space has the same class in H_0 if and only if there exists an…

Let $\psi(G) = \sum_{g \in G} o(g)$ denote the sum of element orders of a finite group $G$. It is known that among groups of order $n$, the cyclic group $C_n$ maximizes $\psi$. T\u{a}rn\u{a}uceanu proved that two finite abelian $p$-groups…

The (proper) power graph of a group is a graph whose vertex set is the set of all (nontrivial) elements of the group and two distinct vertices are adjacent if one is a power of the other. Various kinds of planarity of (proper) power graphs…

We investigate group coding for arbitrary finite groups acting linearly on a vector space. These yield robust codes based on real or complex matrix groups. We give necessary and sufficient conditions for correct subgroup decoding using…

We investigate coherency properties of certain completed integral group rings, precisely for compact $p$-adic Lie groups.

Let $G$ be a finite group. We will say that $M$ and $S$ form a \textsl{complete splitting} (\textsl{splitting}) of $G$ if every element (nonzero element) $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$,…

Let $G$ be a finite group. Then we denote $\psi(G) = \sum_{x\in G}o(x)$ where $o(x)$ is the order of the element $x$ in $G$. In this paper we characterize some finite $p$-groups ($p$ a prime) by $\psi$ and their orders.

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than 1 to exactly one vertex of $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code…

We will explore the nature of when certain finite groups have an equal covering, and when finite groups do not. Not to be confused with the concept of a cover group, a covering of a group is a collection of proper subgroups whose…

Given a construction $f$ on groups, we say that a group $G$ is \textit{$f$-realisable} if there is a group $H$ such that $G\cong f(H)$, and \textit{completely $f$-realisable} if there is a group $H$ such that $G\cong f(H)$ and every…

We define a notion of roundness for finite groups. Roughly speaking, a group is round if one can order its elements in a cycle in such a way that some natural summation operators map this cycle into new cycles containing all the elements of…

We consider Cayley sum graphs over the cyclic group $\mathbb{Z}_n$ and aim to explore several necessary and sufficient conditions for the existence of total perfect codes in these graphs. Specifically, we examine various cases for the…

We introduce a criterion for the completeness of ring approximations of modular cohomology rings of finite non prime power groups, and discuss how this criterion performs in practical computations, compared with other criteria.

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the…

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in…

The class of all quasigroups is covered by six classes: the class of all asymmetric quasigroups and five varieties of quasigroups (commutative, left symmetric, right symmetric, semi-symmetric and totally symmetric). Each of these classes is…

We study equilibrium configurations of infinitely many identical particles on the real line or finitely many particles on the circle, such that the (repelling) force they exert on each other depends only on their distance. The main question…