Related papers: On groups in which every element has a prime power…

The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of…

Let the group $G = AB$ be the product of the subgroups $A$ and $B$. We determine some structural properties of $G$ when the $p$-elements in $A\cup B$ have prime power indices in $G$, for some prime $p$. More generally, we also consider the…

We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either…

A subset $X$ of a finite group $G$ is said to be prime-power-independent if each element in $X$ has prime power order and there is no proper subset $Y$ of $X$ with $\langle Y, \Phi(G)\rangle = \langle X, \Phi(G)\rangle$, where $\Phi(G)$ is…

A finite group $G$ is said to be a $\mathcal{B}_{\psi}$-group if $\psi(H)<|G|$ for any proper subgroup $H$ of $G$, where $\psi(H)$ denotes the sum of element orders of $H$. In this paper, we characterize the $\mathcal{B}_{\psi}$-groups up…

Let $G$ be a finite group. Then we denote $\psi(G) = \sum_{x\in G}o(x)$ where $o(x)$ is the order of the element $x$ in $G$. In this paper we characterize some finite $p$-groups ($p$ a prime) by $\psi$ and their orders.

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…

We study the finite solvable groups $G$ in which every real element has prime power order. We divide our examination into two parts: the case $\textbf{O}_2(G)>1$ and the case $\textbf{O}_2(G)=1$. Specifically we proved that if…

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 $C_{p^nm}$…

A group is called capable if it is a central factor group. For each prime $p$ and positive integer $c$, we prove the existence of a capable $p$-group of class $c$ minimally generated by an element of order $p$ and an element of order…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \Pi $-property in $ G $ if there exists a $G$-chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that $ | G…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \Pi $-property in $ G $ if if there exists a chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that for…

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every…

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…

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…

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

Let $G$ be a finite group and $N(G)$ be the set of conjugacy class sizes of $G$. For a prime $p$, let $|G||_p$ be the highest $p$-power dividing some element of $N(G)$. and define $|G|| = {\Pi}_{p\in {\pi}(G)}|G||_p$. $G$ is said to be an…

A finite group $P$ is said to be \emph{primary} if $|P|=p^{a}$ for some prime $p$. We say a primary subgroup $P$ of a finite group $G$ satisfies the \emph{Frobenius normalizer condition} in $G$ if $N_{G}(P)/C_{G}(P)$ is a $p$-group provided…

A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$\Pi$-property in $G$ if $G$ has a chief series $\Gamma_{G}: 1=G_{0}<G_{1}<\cdots<G_{n}=G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leqslant i\leqslant…

Let G be a locally compact Hausdorff group in which every element is of finite order, and let P(G) denote the class of all regular probability measures on G. In this note, it is observed that a characterization of algebraically regular…