Alladi Ramakrishnan Hall

Products of conjugacy classes in finite groups

Rijubrata Kundu

IMSc

Let $G$ be a finite group. Given conjugacy classes $C_1, C_2,\ldots,C_k$ of $G$, the product of these classes can be defined naturally:
$C_1C_2\cdots C_k:=\{x_1x_2\cdots x_k \mid x_i \in C_i for all i\}$. One of the basic
questions is to estimate how much of the group can be covered by the above
product. In the first part of the talk, we will discuss some results and
important conjectures on this topic (of special importance are the
non-Abelian finite simple groups).

In the second part of the talk, we consider the finite symmetric and
alternating groups. In the symmetric group, we consider powers of the
conjugacy class of cycles of a fixed length and determine conditions under
which they can cover the normal subgroup of all even permutations, that is,
the alternating group. We will provide complete answer to a conjecture of
Herzog, Lev, and Kaplan [Herzog, Marcel; Kaplan, Gil; Lev, Arieh; Covering
the alternating groups by products of cycle classes. J. Combin. Theory Ser.
A 115 (2008), no. 7, 1235–1245]. This part of the talk is joint work with
Harish Kishnani and Dr. Sumit Chandra Mishra.