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 $i\in \{1,2,\cdots ,r\}$, $j\in \{1,2,\cdots ,n_i\}$.

Theorem A yields the condition under which the number of solutions of equation $x^{p^k}=1$ in a finite $p$-group is divisible by $p^{n+k}$ (here $n$ is a fixed positive integer). Theorem B which is due to Avinoam Mann generalizes the counting part of the Sylow Theorem. We show in Theorems C and D that congruences for the number of cyclic subgroups of order $p^k$ which are true for abelian groups hold for more general finite groups (for example for groups with abelian Sylow $p$-subgroups).

In this paper as the main result, we determine finite groups with the same prime graph as the automorphism group of a sporadic simple group, except $J_2$.

Let $G$ be a finite group. The prime graph of $G$ is a simple graph $\Gamma \left(G\right)$ whose vertex set is $\pi \left(G\right)$ and two distinct vertices $p$ and $q$ are joined by an edge if and only if $G$ has an element of order $pq$. A group $G$ is called $k$-recognizable by prime graph if there exist exactly $k$ nonisomorphic groups $H$ satisfying the condition $\Gamma \left(G\right)=\Gamma \left(H\right)$. A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that $\mathrm{PGL}(2,p^{\alpha })$ is recognizable, if $p$ is an odd prime and $\alpha >1$ is odd. But for even $\alpha$, only the recognizability...

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...