Related papers: On Commutation Semigroups of Dihedral Groups

A regular ordered semigroup $S$ is called right inverse if every principal left ideal of $S$ is generated by an $\mathcal{R}$-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular…

Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket;…

An oscillator group $G$ is a semidirect product of a Heisenberg group with a one-parameter group. In this article we construct Olshanski semigroups for infinite-dimensional oscillator groups. These are complex involutive semigroups which…

We present a classification of finite $p$-groups $G$ with $\gamma_2(G)$, the commutator subgroup of $G$, of order $p^4$ and exponent $p$ such that not all elements of $\gamma_2(G)$ are commutators.

We present a characterisation of groups $G$ of order $p^7$, $p$ prime, in which not all elements of the commutator subgroup $\gamma_2(G)$ of $G$ are commutators in $G$. On the way we obtain several structural results on groups of order…

'A semigroup is completely regular if and only if it is a union of groups'- an analogue of this structure theorem of completely regular semigroup has been obtained in the setting of seminearrings in [[16], Mukherjee (Pal) et al., Semigroup…

From any directed graph $E$ one can construct the graph inverse semigroup $G(E)$, whose elements, roughly speaking, correspond to paths in $E$. Wang and Luo showed that the congruence lattice $L(G(E))$ of $G(E)$ is upper-semimodular for…

Let ${\cal K}_1(G)$ denote the inverse subsemigroup of ${\cal K}(G)$ consisting of all right cosets of all non-trivial subgroups of $G$. This paper concentrates on the study of the group $\Sigma({\cal K}_1(G))$ of all units of the…

Let G be a connected, simply connected Poisson-Lie group with quasitriangular Lie bialgebra g. An explicit description of the double D(g) is given, together with the embeddings of g and g^*. This description is then used to provide a…

Let $G$ be a connected complex semisimple Lie group, $\Gamma$ be a cocompact, irreducible and torsionless lattice in $G$ and $K$ be a maximal compact subgroup of $G$. Assume $\Gamma$ acts by left multiplication and $K$ acts by right…

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting…

We classify the pairs $(C,G)$ where $C$ is a seminormal curve over an arbitrary field $k$ and $G$ is a smooth connected algebraic group acting faithfully on $C$ with a dense orbit, and we determine the equivariant Picard group of $C$. We…

Given any group $G$, the normalizer $\mathrm{Hol}(G)$ of the subgroup of left translations in the group of all permutations on $G$ is called the holomorph, and the normalizer $\mathrm{NHol}(G)$ of $\mathrm{Hol}(G)$ in turn is called the…

Let $G$ be a finite group. We consider the set of the irreducible complex characters of $G$, namely $Irr(G)$, and the related degree set $cd(G)=\{\chi(1) : \chi\in Irr(G)\}$. Let $\rho(G)$ be the set of all primes which divide some…

Given a finite nonabelian semisimple group $G$, we describe those groups that have the same holomorph as $G$, that is, those regular subgroups $N\simeq G$ of $S(G)$, the group of permutations on the set $G$, such that…

A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate many novel equations. Two independent methods that can be used to derive the equations of the semigroup are…

Given a finite graph G there is a corresponding group given by the presentation with generators the vertices of G and a relation [x,y]=1 for generators x and y precisely when (x,y) is an edge of G. Such groups are known as partially…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal…

A subsemigroup S of a semigroup Q is a local left order in Q if, for every maximal subgroup H of Q, the intersection of S with H is a local left order in the sense of group theory. That is, every q in H can be written as a#b for some a,b in…

Let $A \leq G$ be a subgroup of a group $G$. An $A$-complement of $G$ is a subgroup $H$ of $G$ such that $G = A H$ and $A \cap H = \{1\}$. The \emph{classifying complements problem} asks for the description and classification of all…