Related papers: On groups with same number of centralizers

A finite group G is called Schur, if every Schur ring over G is associated in a natural way with a regular subgroup of Sym(G) that is isomorphic to G. We prove that any nonabelian Schur group G is metabelian and the number of distinct prime…

Let G be a group. The intersection graph G(G) of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G; and there is an edge between two distinct…

For any group G, let C(G) denote the set of centralizers of G. We say that a group G has n centralizers (G is a Cn-group) if |C(G)| = n. In this note, we prove that every finite Cn-group with n ? 21 is soluble and this estimate is sharp.…

The main theorem in this article shows that a group of odd order which admits the alternating group of degree 5 with an element of order 5 acting fixed point freely is nilpotent of class at most two. For all odd primes r, other than 5, we…

Every finite non-abelian group of order $n$ has a non-central element whose centralizer has order exceeding $n^{1/3}$. The proof does not rely on the classification of finite simple groups, yet it uses the Feit-Thompson theorem.

We consider the following two finiteness conditions on normalizers and centralizers in a group G: (i) |N_G(H):H| is finite for every non-normal subgroup H of G, and (ii) |C_G(x):<x>| is finite for every non-normal cyclic subgroup <x> of G.…

Let $G$ be a finitely generated group and let $C^*$ denote the group of all central automorphisms of $G$ fixing the center of $G$ elementwise. Azhdari and Malayeri [J. Algebra Appl., {\bf 6}(2011), 1283-1290] gave necessary and sufficient…

In this article, we prove that if all non-trivial cyclic subgroups of a group $G$ are self normalizing and $G$ satisfies the implication $$ \ o(x)\neq o(y)\Rightarrow o(xy)\neq o(x), o(y), $$ for all non-trivial elements $x$ and $y$, then…

Let $G$ be a finite abelian group. Ferraz, Guerreiro and Polcino Milies prove that the number of $G$-equivalence classes of minimal abelian codes is equal to the number of $G$-isomorphism classes of subgroups for which corresponding…

A group G is a cn-group if for each subgroup H of G there exists a normal subgroup N of G such that the index of both H and N in HN is finite. The class of cn-groups contains properly the classes of core- finite groups and that of groups in…

Let $G$ and $H$ be groups that act compatibly on each other. We denote by $[G,H]$ the derivative subgroup of $G$ under $H$. We prove that if the set $\{g^{-1}g^h \mid g \in G, h \in H\}$ has $m$ elements, then the derivative $[G,H]$ is…

We describe $\{2,3\}$-groups in which the order of a product of any two elements of orders at most $4$ does not exceed $9$ and the centralizer of every involution is a locally cyclic $2$-subgroup. In particular, we will prove that these…

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p; then G contains a simple group as a subquotient which exhibits the same property. In addition…

Let $G$ be a finite non abelian group. The centralizer graph of $G$ is a simple undirected graph $\Gamma_{cent}(G)$, whose vertices are the proper centralizers of $G$ and two vertices are adjacent if and only if their cardinalities are…

It is known that an abelian group $A$ and a $2$-cocycle $c:A \times A \to C$ yield a group ${\mathscr{H}}(A,C,c)$ which we call a Heisenberg group. This group, a central extension of $A$, is the archetype of a class~$2$ nilpotent group. In…

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite $p$-groups and finite simple groups with…

A group $G$ is said to be factorized into subsets $A_1, A_2, \ldots, A_s\subseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1g_2\ldots g_s$, where $g_i\in A_i$, $i=1,2,\ldots,s$. We consider the following…

Let $p$ be a prime. A $p$-group $G$ is defined to be semi-extraspecial if for every maximal subgroup $N$ in $Z(G)$ the quotient $G/N$ is a an extraspecial group. In addition, we say that $G$ is ultraspecial if $G$ is semi-extraspecial and…

Let $G$ be a group and let $R$ be a $G$-graded ring. We show that a nonzero central idempotent in $R$ has finite support group in two broad settings: when $G$ is abelian, and when $G$ is arbitrary but the grading satisfies a certain…

Suppose $G$ is a finite group. The set of all centralizers of $2-$element subsets of $G$ is denoted by $2-Cent(G)$. A group $G$ is called $(2,n)-$centralizer if $|2-Cent(G)| = n$ and primitive $(2,n)-$centralizer if $|2-Cent(G)| =…