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On the Number of Representations of a Finite Group over a Finite Field

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.



Last modified: 2016/12/21 16:09

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