‎‎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...

Abstract We define isoclinism of skew braces and present several applications. study some properties that are invariant under isoclinism. For example, we prove right nilpotency is an invariant. This result has application in the theory set‐theoretic solutions to Yang–Baxter equation. isoclinic multipermutation

In this paper‎, ‎we first present some properties of lower and upper central series of pair of groups‎. ‎Then the notion of $n$-isoclinism for the classification of pairs of groups is introduced‎, ‎and some of the structural properties of the created classes are proved‎. ‎Moreover some interesting theorems such as Baer Theorem‎, ‎Bioch Theorem‎, ‎Hirsh Theorem for pair of groups are generalized...

‎‎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...

Exterior isoclinism gives an alternative classification method for finite Lie algebras and crossed modules. In this paper, we define the notions of exterior Also, generalize these n-th dimension.

Hom-Lie superalgebras can be considered as the deformation of Lie superalgebras; which are $\mathbb{Z}_2$-graded generalization algebras. The motivation this paper is to introduce concept isoclinism and factor set in regular superalgebras. Finally, we obtain that two finite same dimensional isoclinic if only they isomorphic.