Counting subgroups of finite groups is one of the most important topics in finite group theory. We classify the finite non-nilpotent groups $G$ whose set of numbers of subgroups of possible orders $n(G)$ has exactly two elements. We show that if $G$ is a non-nilpotent group whose set of numbers of subgroups of possible orders has exactly 2 elements, then $G$ has a normal Sylow subgroup of prime order and $G $ is solvable. Moreover, as an application we give a detailed description of non-nilpotent groups with $n(G)=\{1, q+1\}$ for some prime $q$. In particular, $G$ is supersolvable under this condition.
A subgroup $H$ of a group $G$ is said to be complemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K=1$. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about $p$-nilpotent groups.
This paper gives necessary and sufficient conditions for subgroups with trivial core to be of odd depth. We show that a subgroup with trivial core is an odd depth subgroup if and only if certain induced modules from it are faithful. Algebraically this gives a combinatorial condition that has to be satisfied by the subgroups with trivial core in order to be subgroups of a given odd depth. The condition can be expressed as a certain matrix with $\{0,1\}$-entries to have maximal rank. The entries of the matrix correspond to the sizes of the intersections of the subgroup with any of its conjugate.
For a finite group $G$ denote by $N(G)$ the set of conjugacy class sizes of $G$. In 1980s, J. G. Thompson posed the following conjecture: If $L$ is a finite nonabelian simple group, $G$ is a finite group with trivial center and $N(G) = N(L)$, then $G\cong L$. We prove this conjecture for an infinite class of simple groups. Let $p$ be an odd prime. We show that every finite group $G$ with the property $Z(G)=1$ and $N(G) = N(A_i)$ is necessarily isomorphic to $A_i$, where $i\in\{2p,2p+1\}$., Azam Babai, Ali Mahmoudifar., and Obsahuje bibliografii