#### Date Thesis Awarded

5-2018

#### Access Type

Honors Thesis -- Access Restricted On-Campus Only

#### Degree Name

Bachelors of Science (BS)

#### Department

Mathematics

#### Advisor

Christopher Ryan Vinroot

#### Committee Members

Joshua Erlich

Eric Swartz

#### Abstract

An element $g$ in a group $G$ is real if there exists $x\in G$ such that $xgx^{-1}=g^{-1}$. If $g$ is real then all elements in the conjugacy class of $g$ are real. In \cite{GS1} and \cite{GS2}, Gill and Singh showed that the number of real $\mathrm{GL}_n(q)$-conjugacy classes contained in $\mathrm{SL}_n(q)$ equals the number of real $\mathrm{PGL}_n(q)$-conjugacy classes when $q$ is even or $n$ is odd. In this paper, we use generating functions to show that the result is also true for odd $q$. We then follow the methods of \cite{GS1} and \cite{GS2} to count the number of real $\mathrm{U}_n(q)$ conjugacy classes contained in $\mathrm{SU}_n(q)$ and the number of real conjugacy classes in $\mathrm{PGU}_n(q)$, and we show that these are equal to the analogous quantities for $\mathrm{GL}_n(q)$ and $\mathrm{PGL}_n(q)$. Thus, we show that these four sets of conjugacy classes have equal size for all $n,q$.

#### Recommended Citation

Amparo, Elena, "Counting Real Conjugacy Classes in Some Finite Classical Groups" (2018). *Undergraduate Honors Theses.* William & Mary. Paper 1238.

https://scholarworks.wm.edu/honorstheses/1238

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Share Alike 4.0 License.