Related papers: On p-groups of maximal class

In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order $p^n$ and nilpotence class $l$, then $\eta (G)$ is bounded below by a linear function in…

Let $G$ be a finite $p$-group of order $p^n$. YA. G. Berkovich (Journal of Algebra {\bf 144}, 269-272 (1991)) proved that $G$ is elementary abelian $p$-group if and only if the order of its Schur multiplier, $M(G)$, is at the maximum case.…

A group is called capable if it is a central factor group. We consider the capability of finite groups of class two and exponent $p$, $p$ an odd prime. We restate the problem of capability as a problem about linear transformations, which…

It is important to classify covering subgroups of the fundamental group of a topological space using their topological properties in the topologized fundamental group. In this paper, we introduce and study some topologies on the fundamental…

For a $p$-group of order $p^n$, it is known that the order of $2$-nilpotent multiplier is equal to $|\mathcal{M}^{(2)}(G)|=p^{\f12n(n-1)(n-2)+3-s_2(G)}$ for an integer $s_2(G)$. In this article, we characterize all of non abelian $p$-groups…

In this paper we study the finite groups in which every element has prime power order, briefly them EPPO-groups. The classification of EPPO-groups is given including the cases of solvable, non-solvable and simple EPPO-groups. This paper is…

A finite $p$-group $G$ is said to be $d$-maximal if $d(H)<d(G)$ for every subgroup $H<G$, where $d(G)$ denotes the minimal number of generators of $G$. A similar definition can be formulated when $G$ is acted on by some group $A$. We…

Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, $p$-radical subgroups typically play a critical role. To classify the…

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…

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…

Let $G$ be a $p$-group and let $\chi$ be an irreducible character of $G$. The codegree of $\chi$ is given by $|G:\text{ker}(\chi)|/\chi(1)$. This paper investigates the relationship between the nilpotence class of a group and the inclusion…

The aim of this work is to find some exact sequences on the $c$- nilpotent multiplier of a group $G$. We also give an upper bound for the $c$- nilpotent multiplier of finite $p$-groups and give the explicit structure of groups whose take…

We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the…

Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set $\{g\in G \ | \ g^{p^n}=1\}$ is a nontrivial subgroup for some $n$, then $G$ is a…

We study the maximal subgroups of branch groups and obtain a criterion that ensures that certain spinal groups are contained in the class $\mathcal{MF}$ of groups with all maximal subgroups of finite index. This allows us to construct…

We classify, up to conjugacy, the subgroups of the Cremona group isomorphic to (Z/p)^r, where p is prime and r is maximal.

A sharp bound is derived for the nilpotency class of a regular p-group in terms of its coexponent, and is used to show that the number of groups of order p^n with a given fixed coexponent, is independent of n, for p and n sufficiently…

Let $p$ be a prime number and $\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$. We study finite groups with abelian derived subgroup and exponent $p$ in terms of group extension data and their matrix presentations. We show a one-to-one correspondence…

We consider groups of the nilpotency class $3$ of order $p^4$ which are the additive groups of local nearrings. It was shown that, for $p>3$, there exist a local nearring on one of such 4 groups.

The paper explores the effect of powerful class of Sylow $p$-subgroups of a given finite group on control of transfer or fusion. We also find an explicit bound for the $p$-length of a $p$-solvable group in terms of the poweful class of a…