Related papers: Explicit elements of norm one for cyclic groups

Let R be a unitary ring of finite cardinality P^k, where p is a prime number and $p\nmid k$. We show that if the group of units of $R$ has at most one subgroup of order $p$, then $R\cong A\bigoplus B,$ where $B$ is a finite ring of order…

Let $R$ be an associative unital algebra over a field $k,$ let $p$ be an element of $R,$ and let $R'=R\langle q\mid pqp= p\rangle.$ We obtain normal forms for elements of $R',$ and for elements of $R'$-modules arising by extension of…

Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $\vp : RG\to RG$ denote the $R$-linear extension of an involution $\vp$ defined on $G$. An element $x$ in $RG$ is said to be $\vp$-antisymmetric if $\vp (x) = -x$. A…

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.

If $G$ is a group acting geometrically on a CAT(0) cube complex $X$ and if $g \in G$ is an infinite-order element, we show that exactly one of the following situations occurs: (i) $g$ defines a rank-one isometry of $X$; (ii) the stable…

We focus our attention to the set $\gl{\coring{C}}$ of grouplike elements of a coring $\coring{C}$ over a ring $A$. We do some observations on the actions of the groups $U(A)$ and $\aut{\coring{C}}$ of units of $A$ and of automorphisms of…

In this paper we measure how efficiently a finite simple group $G$ is generated by its elements of order $p$, where $p$ is a fixed prime. This measure, known as the $p$-width of $G$, is the minimal $k\in \mathbb{N}$ such that any $g\in G$…

In this paper, we continue the enumeration of Schur rings over cyclic groups. Cyclic groups of semiprime order $pq$, where $p$ and $q$ are distinct primes, are considered. Additionally, cyclic groups of order $4p$ are considered.

We determine the structure of automorphism group or each nonsplit metacyclic 2-group. This completes the work on automorphism groups of metacyclic $p$-groups.

Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American…

The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of…

It is shown in this paper that non-conforming finite elements on the triangle using $P^{1}$-nonconforming polynomials and $P^{2}$ -conforming polynomials can be easily built and used.They appear as an 'enriched' version of the standard…

Let $p$ be a prime number, $F$ a field of characteristic $p$, and $G$ a cyclic group of order $q =p^a$ for some positive integer $a$. Under these circumstances every indecomposable $F G$-module is cyclic. For indecomposable $F G$-modules…

Let $q$ be a power of a prime $p$ and let $G$ be a completely reducible subgroup of $\mathrm{GL}(d,q)$. We prove that the number of composition factors of $G$ that have prime order $p$ is at most $(\varepsilon_q d-1)/(p-1)$, where…

A Schur ring (S-ring) over a group $G$ is called separable if every of its similaritities is induced by isomorphism. We establish a criterion for an S-ring to be separable in the case when the group $G$ is cyclic. Using this criterion, we…

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…

Let the group $G = AB$ be the product of the subgroups $A$ and $B$. We determine some structural properties of $G$ when the $p$-elements in $A\cup B$ have prime power indices in $G$, for some prime $p$. More generally, we also consider the…

Given an irreducible well-generated complex reflection group W with Coxeter number h, we call a Coxeter element any regular element (in the sense of Springer) of order h in W; this is a slight extension of the most common notion of Coxeter…

Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $\phi_{\sigma} : RG\to RG$ denote the involution defined by $\phi_{\sigma} (\sum r_{g}g) = \sum r_{g} \sigma (g) g^{-1}$, where $\sigma:G\to \{\pm 1\}$ is a group…

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged…