Related papers: C-Groups

We define and we characterize regular and c-regular cyclically ordered abelian groups. We prove that every dense c-regular cyclically ordered abelian group is elementarily equivalent to some cyclically ordered group of unimodular complex…

Many common finite p-groups admit automorphisms of order coprime to p, and when p is odd, it is reasonably difficult to find finite p-groups whose automorphism group is a p-group. Yet the goal of this paper is to prove that the automorphism…

Suppose that $\mathcal{C}$ is a class of groups consisting only of periodic groups and $\mathfrak{P}(\mathcal{C})^{\prime}$ is the set of prime numbers each of which does not divide the order of any element of a $\mathcal{C}$-group. A…

Let $G$ be a finite group, define $I(G)=\{x\in G : x^{2}=1\}$, $C(G)=$ set of the cyclic subgroups of $G$, $i(G)=|I(G)|$ and $c(G)=|C(G)|$. In this article, we will classify finite groups with $i(G)=c(G)-r$ for $r=0,1,$ and $2$. We also…

For any group G, let C(G) denote the intersection of the normal- izers of centralizers of all elements of G. Set C0 = 1. Define Ci+1(G)=Ci(G) = C(G=Ci(G)) for i ? 0. By C1(G) denote the terminal term of the ascending series. In this paper,…

We investigate structural properties of non-sofic groups, assuming that such groups exist. We introduce and study two classes: minimal non-sofic groups and $\omega$-non-sofic groups. For minimal non-sofic groups, we establish strong…

We study finite p-groups G of coclass upto 4 for which the group Aut_z(G) of all central automorphisms of G is of minimal possible order. As a consequence, we obtain very short and elementary proofs of main results of Sharma and Gumber [7].

A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups…

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\geq 2$, let $C_n$…

Following Isaacs (see [Isa08, p. 94]), we call a normal subgroup N of a finite group G large, if $C_G(N) \leq N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing…

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…

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and…

Groups of finite type (also called finitely constrained groups), introduced by Grigorchuk, are known to be the closure of regular branch groups. This article explores many of their properties. Firstly, we prove that being finitely…

The Gruenberg-Kegel graph $\Gamma(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ if and only if $G$ contains an element of order $pq$. This graph has been the subject of much…

The class of elementary totally disconnected groups is the smallest class of totally disconnected, locally compact, second countable groups which contains all discrete countable groups, all metrizable pro-finite groups, and is closed under…

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group…

We establish several new bounds for the number of conjugacy classes of a finite group, all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group. To apply…

A subgroup $H$ is said to be $nc$-supplemented in a group $G$ if there is a subgroup $K\leq G$ such that $HK\unlhd G$ and $H\cap K$ is contained in $H_G$, the core of $H$ in $G$. We characterize the solvability of finite groups $G$ with…

In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each…

We consider the group property of being icc. We give several examples of icc groups and study its stability under usual algebraic constructions.