Related papers: The derived series of a finite p-group

More than 70 years ago, P. Hall showed that if $G$ is a finite $p$-group such that a term $\der G{d+1}$ of the derived series is non-trivial, then the order of the quotient $\der Gd/\der G{d+1}$ is at least $p^{2^d+1}$. Recently Mann proved…

In this paper we classify all capable finite $p$-groups with derived subgroup of order $p$ and $G/G'$ of rank $n-1$.

Let $d(G)$ be the minimum number of elements required to generated a group $G.$ For a group $G $ of order $p^n$ with derived subgroup of order $ p^k $ and $d(G) = d,$ we knew the order of the Schur multiplier of $G$ is bounded by $…

Let $G$ be a finite $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. In this article we classify non-abelian $p$-groups $G$ of…

We study the group of ends of a pro-p group G and prove a pro-p analog of Stallings' decomposition theorem.

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on…

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal…

Let $G$ be a finite group. The group pseudo-algebra of $G$ is defined as the multi-set $C(G)=\{(d,m_G(d))\mid d\in{\rm Cod}(G)\},$ where $m_G(d)$ is the number of irreducible characters of with codegree $d\in {\rm Cod}(G)$. We show that…

Given a finite group $G$ of order $p^nm$, where $p$ is a prime and $p\nmid m$, we denote by $\psi_p(G)$ the sum of orders of $p$-parts of elements in $G$. In the current note, we prove that $\psi_p(G)\leq\psi_p(C_{p^nm})$, where $C_{p^nm}$…

Let $G$ be a non-abelian $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)| \leq p^{\frac{1}{2}n(n-1)}$. So $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. The groups has…

Let $G$ be a finite $p$-group of order $p^n$. It is known that $|\mathcal{M}(G)|=p^{\f{1}{2}n(n-1)-t(G)}$ and $t(G)\geq 0$. The structure of $G$ characterized when $t(G)\leq 4$ in \cite{be,el,ni,sa,zh}. The structure description of $G$ is…

We prove that $d(G) \log |G| = O(n^2 \log q)$ for irreducible subgroups $G$ of GL$(n,q)$, and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various…

In this note, we give short proofs of the well-known results that the exponent of the Schur multiplier $\M$ divides the exponent of $\G$ for finite $\p$-groups of maximal class and potent $\p$-groups. Moreover, we prove the same for a…

In this short note we confirm a conjecture of James Wiegold. We prove that if $G$ is a finite $p$-group and $|G'|>p^{n(n-1)/2}$ for some non-negative integer $n$, then the group $G$ can be generated by the elements of breadth at least $n$.…

We present some congruences modulo $p^{6-d}$ for sums of the type $\sum_{k=0}^{(p-3)/2}x^k{2k\choose k}/(2k+1)^d$, for $d=1,2,3$ where $p>5$ is a prime.

Recently Rai obtained an upper bound for the order of the Schur multiplier of a $d$-generator special $p$-group when its derived subgroup has the maximum value $ p^{\frac{1}{2}d(d-1)}$ for $ d\geq 3 $ and $ p\neq 2. $ Here we try to obtain…

A famous result of P. Heged\"us describes the structure of finite rational $\{2,5\}$-groups. In this paper, we show how Heged\"us' proof can be used to obtain a similar result for the more general situation of a finite rational $2$-group…

Let $G$ be a finite group of order divisible by a prime $p$ and let $P\in\Syl_p(G)$. We prove a recent conjecture by Hung stating that $|\Irr_{p'}(G)|\geq \frac{\exp(P/P')-1}{p-1}+2\sqrt{p-1}-1.$ Let $a\geq 2$ be an integer and suppose that…

In this paper we study the ratio between the number of $p$-elements and the order of a Sylow $p$-subgroup of a finite group $G$. As well known, this ratio is a positive integer and we conjecture that, for every group $G$, it is at least the…

We prove that any finitely generated one ended group has linear end depth. Moreover, we give alternative proofs to theorems relating the growth of a finitely generated group to the number of its ends.