We investigate the situation when the inner mapping group of a commutative loop is of order $2p$, where $p=4t+3$ is a prime number, and we show that then the loop is solvable.

Loop capable groups are groups which are isomorphic to inner mapping groups of loops. In this paper we show that abelian groups $C_p^k\times C_p\times C_p$, where $k\ge 2$ and $p$ is an odd prime, are not loop capable groups. We also discuss generalizations of this result.

In this paper we consider finite loops and discuss the problem which nilpotent groups are isomorphic to the inner mapping group of a loop. We recall some earlier results and by using connected transversals we transform the problem into a group theoretical one. We will get some new answers as we show that a nilpotent group having either $C_{p^k}\times C_{p^l}$, $k>l\ge 0$ as the Sylow $p$-subgroup for some odd prime $p$ or the group of quaternions as the Sylow $2$-subgroup may not be loop capable.

Let $G$ be a group. A subgroup $H$ of $G$ is called a TI-subgroup if $H\cap H^g=1$ or $H$ for every $g\in G$ and $H$ is called a QTI-subgroup if $C_G\left(x\right)\le N_G\left(H\right)$ for any $1\ne x\in H$. In this paper, a finite group in which every nonabelian maximal is a TI-subgroup (QTI-subgroup) is characterized.

Suppose $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is said to be $s$-permutably embedded in $G$ if for each prime $p$ dividing $\left|H\right|$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-permutable subgroup of $G$; $H$ is called weakly $s$-permutably embedded in $G$ if there are a subnormal subgroup $T$ of $G$ and an $s$-permutably embedded subgroup $H_{se}$ of $G$ contained in $H$ such that $G=HT$ and $H\cap T\le H_{se}$. We investigate the influence of weakly $s$-permutably embedded subgroups on the $p$-nilpotency and $p$-supersolvability of finite...

A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$. In this paper, some interesting results with weakly-supplemented minimal subgroups or Sylow subgroups of $G$ are obtained.

A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$. In this paper, some interesting results with weakly-supplemented minimal subgroups to a smaller subgroup of $G$ are obtained.

For a finite group $G$ and a non-linear irreducible complex character $\chi$ of $G$ write $\upsilon \left(\chi \right)=\{g\in G\mid \chi (g)=0\}$. In this paper, we study the finite non-solvable groups $G$ such that $\upsilon \left(\chi \right)$ consists of at most two conjugacy classes for all but one of the non-linear irreducible characters $\chi$ of $G$. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable $\varphi$-groups. As a corollary, we answer Research Problem $2$ in [Y. Berkovich and L. Kazarin: Finite...

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the $\Pi$-property in $G$ if for any chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi (HK/K\cap L/K)$-number. We study the influence of some $p$-subgroups of $G$ satisfying the $\Pi$-property on the structure of $G$, and generalize some known results.

Let $\sigma =\{{\sigma }_i:i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ be a finite group and $\sigma \left(G\right)=\{{\sigma }_i:{\sigma }_i\cap \pi \left(G\right)\ne \varnothing \}$. A set $\mathscr{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every non-identity member of $\mathscr{H}$ is a Hall ${\sigma }_i$-subgroup of $G$ and $\mathscr{H}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma \left(G\right)$. $G$ is said to be $\sigma$-full if $G$ possesses a complete Hall $\sigma$-set. A subgroup $H$ of $G$ is $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathscr{H}$ such that $H{A}^x$= $A^{x}H$ for all $A\in \mathscr{H}$ and all $x\in G$. A subgroup $H$ of $G$ is $\sigma$-permutably embedded in $G$ if $H$ is $\sigma$-full...

We introduce a new subgroup embedding property of finite groups called CSQ-normality of subgroups. Using this subgroup property, we determine the structure of finite groups with some CSQ-normal subgroups of Sylow subgroups. As an application of our results, some recent results are generalized.