The paper reports the results of a search for pairs of groups of order $n$ that can be placed in the distance $n^2/4$ for the case when $n\in \{16,32\}$. The constructions that are used are of the general character and some of their properties are discussed as well.

Let $G$ be a finite group. The prime graph of $G$ is a graph whose vertex set is the set of prime divisors of $\left|G\right|$ and two distinct primes $p$ and $q$ are joined by an edge, whenever $G$ contains an element of order $pq$. The prime graph of $G$ is denoted by $\Gamma \left(G\right)$. It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if $G$ is a finite group such that $\Gamma \left(G\right)=\Gamma \left(B_n\left(5\right)\right)$, where $n\ge 6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_n\left(5\right)$ or $C_n\left(5\right)$.

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).