finite groups with three relative commutativity degrees

Finite groups with three relative commutativity degrees

‎‎For a finite group $G$ and a subgroup $H$ of $G$‎, ‎the relative commutativity degree of $H$ in $G$‎, ‎denoted by $d(H,G)$‎, ‎is the probability that an element of $H$ commutes with an element of $G$‎. ‎Let $mathcal{D}(G)={d(H,G):Hleq G}$ be the set of all relative commutativity degrees of subgroups of $G$‎. ‎It is shown that a finite group $G$ admits three relative commutativity degrees if a...

Commutativity Degrees of Wreath Products of Finite Abelian Groups

We compute commutativity degrees of wreath products A o B of finite abelian groups A and B . When B is fixed of order n the asymptotic commutativity degree of such wreath products is 1/n2. This answers a generalized version of a question posed by P. Lescot. As byproducts of our formula we compute the number of conjugacy classes in such wreath products, and obtain an interesting elementary numbe...

On Finite Groups With Two Irreducible Character Degrees

restrictions on commutativity ratios in finite groups

‎we consider two commutativity ratios $pr(g)$ and $f(g)$ in a finite group $g$‎ ‎and examine the properties of $g$ when these ratios are `large'‎. ‎we show that‎ ‎if $pr(g) > frac{7}{24}$‎, ‎then $g$ is metabelian and we give threshold‎ ‎results in the cases where $g$ is insoluble and $g'$ is nilpotent‎. ‎we also‎ ‎show that if $f(g) > frac{1}{2}$‎, ‎then $f(g) = frac{n+1}{2n}$‎, ‎for some‎ ‎na...

on finite groups with two irreducible character degrees

Relative n-th non-commuting graphs of finite groups

‎Suppose $n$ is a fixed positive integer‎. ‎We introduce the relative n-th non-commuting graph $Gamma^{n} _{H,G}$‎, ‎associated to the non-abelian subgroup $H$ of group $G$‎. ‎The vertex set is $Gsetminus C^n_{H,G}$ in which $C^n_{H,G} = {xin G‎ : ‎[x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin H}$‎. ‎Moreover‎, ‎${x,y}$ is an edge if $x$ or $y$ belong to $H$ and $xy^{n}eq y^{n}x$ or $x...