Kaur, Dilpreet

For a finite group G, we consider the problem of counting simultaneous conjugacy classes of n-tuples and simultaneous conjugacy classes of commuting n-tuples in G. Let αG,n denote the number of simultaneous conjugacy classes of n-tuples, and βG,n the number of simultaneous conjugacy classes of commuting n-tuples in G. The generating functions AG(t) = ∑n≥0 αG,ntn, and BG(t) = ∑n≥0 βG,ntn are rational functions of t. In this paper we study normalized functions AG(t/|G|) and BG(t/|G|) for finite p-groups of rank at most 5.

Let p be a prime number. A p-group is said to be extraspecial p-group if its center, derived subgroup and Frattini subgroup all coincide and have order p. For every n ∈ N, there are exactly two extraspecial p-groups of order p2n+1 up to isomorphism and no extraspecial p-group of order p2n. This classification of extraspecial p-group is proved by using theory of quadratic forms and bilinear forms.

Let K = R or C and Tn denote the Takasaki quandle of order n. In this paper, we provide decomposition of quandle ring K[Tn] as direct sum of right simple ideals. This decomposition is equivalent to decomposition of regular representation [M. Elhamdadi and E. M. Moutuou, Finitely stable racks and rack representations, Comm. Algebra 46(11) (2018) 4787-4802] of Takasaki quandles.

Let K=R or C and Rn be the dihedral quandle of order n. In this article, we give decomposition of the quandle ring K[Rn] into indecomposable right K[Rn]-modules for all even n∈N. It follows that the decomposition of K[Rn] given in [2, Prop. 4.18(2)] is valid only in the case when n is not divisible by 4.

This paper concerns the enumeration of simultaneous conjugacy classes of k-tuples of commuting matrices in the upper triangular group Tn(Fq) and unitriangular group UTm(Fq) over the finite field Fq of odd characteristic. This is done for n = 2, 3, 4 and m = 3, 4, 5, by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities cpk for k ≤ 5 in each case.