restrictions on commutativity ratios in finite groups

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...

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...

Further restrictions on the structure of finite CI-groups

A group G is called a CI-group if, for any subsets S, T ⊂ G, whenever two Cayley graphs Cay(G, S) and Cay(G, T ) are isomorphic, there exists an element σ ∈ Aut(G) such that S = T . The problem of seeking finite CI-groups is a longstanding open problem in the area of Cayley graphs. This paper contributes towards a complete classification of finite CI-groups. First it is shown that the Frobenius...

On Commutativity and Finiteness in Groups

The second author introduced notions of weak permutablity and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite subgroups. Two groups H, K weakly commute provided there exists a bijection f : H → K which fixes the identity and such that h commutes with its image h for all h ∈ H. The present paper gives support to conjectures about the ni...

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...

A Generalization of Commutativity Degree of Finite Groups