p-groups may be classified by splitting the groups up into classes having the same commutator relations (isoclinism classes) and then determining the nonisomorphic groups in each class. This paper reduces the problem of determining the isomorphism classes to that of finding the equivalence classes of a set of matrices under some equivalence relation. A computer is used to find the equivalence c...

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...

We give a sufficient condition on a finite p-group G of nilpotency class 2 so that Autc(G) = Inn(G), where Autc(G) and Inn(G) denote the group of all class preserving automorphisms and inner automorphisms of G respectively. Next we prove that if G and H are two isoclinic finite groups (in the sense of P. Hall), then Autc(G) ∼= Autc(H). Finally we study class preserving automorphisms of groups o...

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...

In the work of Rostami et al., Bogomolov multiplier a Lie algebra L over field Ω is defined as particular factor subalgebra exterior product L∧L. If finite dimensional, we identify this object certain subgroup second cohomology group H2(L,Ω) by deriving Hopf-Type formula. As an application, affirmatively answer two questions posed Kunyavskiĭ regarding invariance under isoclinism algebras and ex...

Let G be a finite group, and let κ(G) be the probability that elements g, h ∈ G are conjugate, when g and h are chosen independently and uniformly at random. The paper classifies those groups G such that κ(G) ≥ 1/4, and shows that G is abelian whenever κ(G)|G| < 7/4. It is also shown that κ(G)|G| depends only on the isoclinism class of G. Specialising to the symmetric group Sn, the paper shows ...

‎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...