#### 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.

Done