Let $G$ be a finite group. A normal subgroup $N$ of $G$ is a union of several $G$-conjugacy classes, and it is called $n$-decomposable in $G$ if it is a union of $n$ distinct $G$-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its...

A group $G$ is said to be a PC-group, if $G/C_G\left(x^G\right)$ is a polycyclic-by-finite group for all $x\in G$. A minimal non-PC-group is a group which is not a PC-group but all of whose proper subgroups are PC-groups. Our main result is that a minimal non-PC-group having a non-trivial finite factor group is a finite cyclic extension of a divisible abelian group of finite rank.

In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group $\mathbf{G}$ defined over a finite field with corresponding Frobenius map $F$ and derive the number of $F$-stable semisimple classes of $\mathbf{G}$ with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters of the fixed point subgroup ${\mathbf{G}}^F$ to their inertia group in the full automorphism group. As a consequence, we...

Let $N$ be a normal subgroup of a group $G$. The structure of $N$ is given when the $G$-conjugacy class sizes of $N$ is a set of a special kind. In fact, we give the structure of a normal subgroup $N$ under the assumption that the set of $G$-conjugacy class sizes of $N$ is $(p_{1n_1}^{a_{1n_1}},\cdots ,p_{11}^{a_{11}},1)\times \cdots \times (p_{r{n}_r}^{a_{r{n}_r}},\cdots ,p_{r1}^{a_{r1}},1)$, where $r>1$, $n_i>1$ and $p_{ij}$ are distinct primes for $i\in \{1,2,\cdots ,r\}$, $j\in \{1,2,\cdots ,n_i\}$.

Gli autori studiano il sottogruppo $\mathrm{\Sigma }\left(G\right)$ intersezione dei sottogruppi massimali e non supersolubili di un gruppo finito $G$ e le relazioni tra la struttura di $G$ e quella di $\mathrm{\Sigma }\left(G\right)$.

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...