#### IMSc Webinar

#### Generating function for the powers in $\text{GL}(n,q)$

#### Rijubrata Kundu

##### IISER Pune

*Zoom link: us02web.zoom.us/j/86865846431*

Let $M\geq 2$ be any integer. Consider the set

$\text{GL}(n,q)^M=\{x^M|x\in \text{GL}(n,q)\}$, which

is the set of all $M^{th}$ powers in the group $\text{GL}(n,q)$. In this talk, we will obtain generating functions for

(a) the proportion of regular and regular semsimple elements in

$\text{GL}(n,q)^M$, assuming $(M,q)=1$,

(b) the proportion of semisimple and all elements which are $M^{th}$ powers when $(M,q)=1$, and $M$ is a power of a prime.

Time permitting we will also discuss the other extreme, where we assume $M$ is a prime and $q$ is a power of $M$.

This is a joint work with Dr. Anupam Singh.

Done