Related papers: R-harmonious groups

Let $\mathscr{R}$ be a type $II_1$ von Neumann algebra. We show that every unitary in $\mathscr{R}$ may be decomposed as the product of six symmetries (that is, self-adjoint unitaries) in $\mathscr{R}$, and every unitary in $\mathscr{R}$…

Let $H$ be a permutation group on a set $\Lambda$, which is permutationally isomorphic to a finite alternating or symmetric group $A_n$ or $S_n$ acting on the $k$-element subsets of points from $\{1,\ldots,n\}$, for some arbitrary but fixed…

A group $G$ is said to be cut if, for every $g \in G$, each generator of $< \! g \! >$ is conjugated to either $g$ or $g^{-1}$. It is conjectured that a Sylow 3-subgroup $P$ of a cut group $G$ is cut. We prove that this is true if $|G|$ is…

For a finite group $G$, let $\omega(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $\omega(H)=\omega(G)$. We say that the recognition problem is solved for $G$ if the…

If $G$ is a finite group and $x\in G$ then the set of all elements of $G$ having the same order as $x$ is called {\em an order subset of $G$ determined by $x$} (see [2]). We say that $G$ is a {\em group with perfect order subsets} or…

In this paper, we prove that, given any integers $d$, $e$, $r$ and $r'$, and a prime $p$ not dividing $de$, any two blocks of the complex reflection groups $G(de,e,r)$ and $G(de,e,r')$ with the same $p$-weight are perfectly isometric.

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $[g_{ij}]$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a…

Two finite groups $L_1$ and $L_2$ are compatible if there exists a finite group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$. We prove a new sufficient condition for two groups to be…

We construct harmonic Riemannian submersions that are retractions from symmetric spaces of noncompact type onto their rank-one totally geodesic subspaces. Among the consequences, we prove the existence of a non-constant, globally defined…

Let $M$ be a closed, connected, orientable topological four-manifold with $H_1(M)$ nontrivial and free abelian, $b_2(M)\ne 0, 2$, and $\chi(M)\ne 0$. We show that if $G$ is a finite group of 2-rank $\le 1$ which admits a homologically…

Many finite groups, including all finite non-abelian simple groups, can be symmetrically generated by involutions. In this paper we give an algorithm to symmetrically represent elements of finite groups and to transform symmetrically…

A complex irreducible character of a finite group G is said to be p-constant, for some prime p dividing the order of G, if it takes constant value at the set of p-singular elements of G. In this paper we classify irreducible p-constant…

A linear group G on a finite vector space V, (that is, a subgroup of GL(V)) is called (1/2)-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the (1/2)-transitive linear…

We study the groups $G$ with the curious property that there exists an element $k\in G$ and a function $f\colon G\to G$ such that $f(xk)=xf(x)$ holds for all $x\in G$. This property arose from the study of near-rings and input-output…

Let $\Gamma(G)$ be the prime graph associated with a finite group $G$ and $D(G)$ be the degree pattern of $G$. A finite group $G$ is said to be $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ such that…

In this paper, we start by giving the definitions and basic facts about hyperoctahedral number system. There is a natural correspondence between the integers expressed in the latter and the elements of the hyperoctahedral group when we use…

Let $G$ be a finite group and $D_{2n}$ be the dihedral group of $2n$ elements. For a positive integer $d$, let $\mathsf{s}_{d\mathbb{N}}(G)$ denote the smallest integer $\ell\in \mathbb{N}_0\cup \{+\infty\}$ such that every sequence $S$…

Motivated by a problem from behavioral economics, we study subgroups of permutation groups that have a certain strong symmetry. Given a fixed permutation, consider the set of all permutations with disjoint inversion sets. The group is…

A finite group $G$ is called a Schur group if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. It is proved that the group $C_3\times C_3\times C_p$ is Schur for any…

A permutation group is said to be quasiregular if every its transitive constituent is regular, and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group: the transitive constituents are replaced by…