International Journal of Group Theory
Volume:2 Issue: 2, Jun 2013

‎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 $G$ thecharacter $alpha chi$ 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 the 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 $psi (G) =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 nonsimple group $G$ of order 60. Also we prove that $psi (PSL(2،7))groups $G$ of order 168. These two results confirm the conjecture posed in [2] for simple groups $A_5$ and $PSL (2،7) $.

Three infinite families of finite abeliab groups will be described such that each members of these families has the Redei 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 ''ll prove that if $G$ is a finite group with $nabla(G)congnabla(BS_{n})$, then $G cong BS_{n}$, where $BS_{n}$ is the symmetric group of degree $n$, where $n$ is a natural number.