The number of factorizations of a finite abelian group as the product of two subgroups is computed in two different ways and a combinatorial identity involving Gaussian binomial coefficients is presented.

The number of factorizations of a finite abelian group as the product of two subgroups is computed in two different ways and a combinatorial identity involving Gaussian binomial coefficients is presented.

We say that a finite group $G$ is conjugacy expansive if for any normal subset $S$ and any conjugacy class $C$ of $G$ the normal set $SC$ consists of at least as many conjugacy classes of $G$ as $S$ does. Halasi, Mar'oti, Sidki, Bezerra have shown that a group is conjugacy expansive if and only if it is a direct product of conjugacy expansive simple or abelian groups. By considering a character analogue of the above, we say that a finite group $G$ is character expansive if for any complex character $alpha$ and irreducible character $chi$ of $G$ the character $alpha chi$ has at least as many irreducible constituents, counting without multiplicity, as $alpha$ does. In this paper we take some initial steps in determining character expansive groups.

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Suppose that ${rm{Irr}} (G | N)$ is the set of the irreducible characters of $G$ that contain $N$ in their kernels. In this paper, we classify solvable groups $G$ in which the set $mathcal{C} (G) = {{rm{Irr}} (G | N) | 1 ne N trianglelefteq G }$ has at most three elements. We also compute the set $mathcal{C}(G)$ for such groups.

In this paper we present some results about subgroup which is generalization of the subgroup $R_{2}^{otimes}(G)={ain G|[a,g]otimes g=1_{otimes},forall gin G}$ of right $2_{otimes}$-Engel elements of a given group $G$. If $p$ is an odd prime, then with the help of these results, we obtain some results about tensor squares of p-groups satisfying the law $[x,g,y]otimes g=1_{otimes}$, for all $x, g, yin G$. In particular p-groups satisfying the law $[x,g,y]otimes g=1_{otimes}$ have abelian tensor squares. Moreover, we can determine tensor squares of two-generator p-groups of class three satisfying the law $[x,g,y]otimes g=1_{otimes}$.

Let $G$ be a finite group. We denote by $psi(G)$ the integer $sum_{gin G}o(g)$, where $o(g)$ denotes the order of $g in G$. Here we show that $psi(A_5)< psi(G)$ for every non-simple group $G$ of order $60$, where $A_5$ is the alternating group of degree $5$. Also we prove that $psi(PSL(2,7))<psi(G)$ for all non-simple groups $G$ of order $168$. These two results confirm the conjecture posed in [J. Algebra Appl., {bf 10} No. 2 (2011) 187-190] for simple groups $A_5$ and $PSL(2,7)$.

Three infinite families of finite abelian groups will be described such that each member of these families has the R'edei $k$-property for many non-trivial values of $k$.

The non-commuting graph $nabla(G)$ of a non-abelian group $G$ is defined as follows: its vertex set is $G-Z(G)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. In this paper we prove that if $G$ is a finite group with $nabla(G) cong nabla(BS_n)$, then $G cong BS_n$, where $BS_n$ is the symmetric group of degree $n$, where $n$ is a natural number.