Let $ \text{GL}_2(\mathbb{F}_5) $ denote the group of $2 \times 2$ invertible matrices over $ \mathbb{F}_5 $, and $ \mathcal{S}_n $ represent the group of permutations of $ n $ objects. Determine the smallest $ n \in \mathbb{N} $ for which there exists an injective homomorphism from $ \text{GL}_2(\mathbb{F}_5) $ to $ \mathcal{S}_n $
My small idea
I haven't solved it yet but let's show $n\ge 11$. Suppose $f:G\hookrightarrow S_n$.
- We have $|G|=(5^2-1)(5^2-5)=480$. Since $f(G)$ is a subgroup of $S_n$ we have $2^5\cdot 3\cdot 5|n! \implies n\ge 8$. Look at the factors 2.
- every element $A\in G$ is diagable.