International Journal of Group Theory
Volume:13 Issue: 2, Jun 2024

Let $R$ be a commutative ring with unity and let $n\geq 3$ be an integer. Let $SL_n(R)$ and $E_n(R)$ denote respectively the special linear group and elementary subgroup of the general linear group $GL_n(R).$ A result of Hurwitz says that the special linear group of size atleast three over the ring of integers of an algebraic number field is finitely generated. A celebrated theorem in group theory states that finite simple groups are two-generated. Since the special linear group of size atleast three over the ring of integers is not a finite simple group, we expect that it has more than two generators. In the special case, where $R$ is the ring of integers of an algebraic number field which is not totally imaginary, we provide for $E_n(R)$ (and hence $SL_n(R)$) a set of Gow-Tamburini matrix generators, depending on the minimal number of generators of $R$ as a $Z$-module.

Poset-blowdown of subgroup posets of groups is an analog of blowdown in algebraic geometry. It is a poset map obtained by contracting normal subgroups. For finite groups, this is considered as a map between the Hasse diagrams of the subgroup posets. Poset-blowdowns are classified into three types: \textit{tame, wild}, and \textit{hybrid} depending on the sizes of their fibers. In this paper we describe the poset-blowdowns for generalized quaternion groups $Q_{2^n}$ $(n \geq 3)$. They have distinguished nature in that all types (tame, wild, and hybrid) appear in the successive poset-blowdowns associated with the three chief series of $Q_{2^n}$.

A simple graph is called semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let $p$ be an arbitrary prime. Folkman [J. Folkman, Regular line-symmetric graphs, J. Combinatorial Theory, \textbf{3} (1967) 215--232.] proved that there are no cubic semisymmetric graphs of order $2p$ or $2p^{2}$. In this paper, an extension of his result in the case of cubic graphs of order $44p$ or $44p^{2}$ is given. By using group theoretic methods, we prove that there are no connected cubic semisymmetric graphs of order $44p$ or $44p^{2}$.

A finite non-Dedekind group $G$ is called an 𝒩𝒜𝒞-group if all non-normal abelian subgroups are cyclic. In this paper, all finite 𝒩𝒜𝒞-groups will be characterized. Also, it will be shown that the center of non-nilpotent 𝒩𝒜𝒞- groups is cyclic. If 𝒩𝒜𝒞-group $G$ has a non-abelian non-normal Sylow subgroup of odd order, then other Sylow subgroups of $G$ are cyclic or of quaternion type.

Let $\mathbb{F}_qG$ be a finite group algebra. We denote by $P(\mathbb{F}_qG)$ the probability that the product of two elements of $\mathbb{F}_qG$ be zero. In this paper, we obtain several results on this probability including a computing formula and characterizations. In particular, the computing formula for the $P(\mathbb{F}_qG)$ are established where $G$ is the cyclic group $C_n$, the Quaternion group $Q_8$, the symmetric group $S_3$ and $F_q$ is a finite field.

We obtain an exact formula for the average order of elements of regular wreath product of two finite groups. Then focussing our attention on $p$-groups for primes $p$, we give an estimate for the verage order of a wreath product $A\wr B$ in terms of maximum order of elements of $A$ and average order of $B$ and an exact formula for the distribution of orders of elements of $A\wr B.$ Finally, we show how wreath products can be used to find several rational numbers which are limits of average orders of a sequence of $p$-groups with cardinalities going to infinity.

Let $c(G)$ denotes the number of cyclic subgroups of a finite group $G$. A group $G$ is said to be {\em $n$-cyclic}, if $c(G)=n$. In this paper, we classify all $11$-cyclic groups.

In this work we present a systematic study of $n$-layered modules which are closely related to $\Sigma$-modules. For each integer $n \geq 1$ we prove some results for $n$-layered modules concerning when $\Sigma$-modules are direct sum of countably generated modules. Moreover, we discover additional restriction which leads to coinciding of $n$-layered modules and $m$-layered modules for $n>m$.