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...

‎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...

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

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.

We consider the problem of counting simultaneous conjugacy classes n-tuples in a finite group G. Let αG,n denote number n-tuples, and βG,n commuting study generating functions AG(t)=∑n≥0αG,ntn, BG(t)=∑n≥0βG,ntn, which are rational t. show that AG(t) determines is completely determined by class equation prove normalized AG(t/|G|) BG(t/|G|) invariants isoclinism families. compute exponential grow...

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups, see Theorem 4.4. Moreover, we show that the stable cohomology of the n-fold wreath product Gn = Z/p o · · · oZ/p of cyclic groups Z/p is detected by elementary abelian p-subgroups and we describe the resulting coho...

a famous theorem of schur states that for a group $g$ finiteness of $g/z(g)$‎ ‎implies the finiteness of $g'.$ the converse of schur's theorem is an interesting problem which has been considered by some‎ ‎authors‎. ‎recently‎, ‎podoski and szegedy proved the truth of the converse of schur's theorem for capable groups‎. ‎they also established an explicit bound for the index of the center of such...