Pornrat Ruengrot
Mahidol University International College

Let $G$ be a finite group, $F$ a field. Recall that a representation of $G$, of degree $n$, over $F$ is a homomorphism $\rho\in Hom(G,GL_n(F))$. When $F$ is a finite field $\mathbb{F}_q$ with $q$ elements, we usually denote $GL_n(F)$ by $GL_n(q)$. In this case, $|Hom(G,GL_n(q))|$ is finite.

In the settings of this talk, we will work with a finite group $G$ and a finite field $F=\mathbb{F}_q$ whose characteristic does not divide $|G|$. Then the celebrated Maschke's theorem implies that every $FG$-module is a direct sum of simple $FG$-submodules. Using this theorem together with a few other results, we will derive a formula for $|Hom(G,GL_n(q))|$.

Finally, we will discuss the case where $G$ is a cyclic group $C_k$ of order $k$. Here, $|Hom(G,GL_n(q))|$ is just the number of ways of sending a generator of $G$ to a matrix $A\in GL_n(q)$ such that $A^k=I$. In other words, it is equal to the number of solutions of $A^k=I$ for $A\in GL_n(q)$. For example, let $a_n$ be the number of such solutions:

• When $k=5, q=4$, the sequence $(a_n)_{n=1}^\infty$ is A053860 in OEIS.
• When $k=10, q=3$, the sequence $(a_n)_{n=1}^\infty$ is A053855 in OEIS.

• Slides (mostly quoted theorems)
• The first 200 terms from the sequence $a_n=|\{X\in GL_n(3): X^2=I\}|$ (cf. A053846).

References

• Kent E. Morrison, Integer sequences and matrices over finite fields - https://arxiv.org/abs/math/0606056
• N. Chigira, Y. Takegahara and T. Yoshida, On the Number of Homomorphisms from a Finite Group to a General Linear Group - http://www.sciencedirect.com/science/article/pii/S0021869399983989
• M.W. Liebeck and A. Shalev, The number of homomorphisms from a finite group to a general linear group - http://wwwf.imperial.ac.uk/~mwl/homjan02.pdf