 Automorphism-count bound for a finite field extension

If $K/k$ is finite, then any distinct $k$-automorphisms ${\sigma }_1,\dots ,{\sigma }_m$ of $K$ are linearly independent in the $K$-vector space ${\mathrm{Hom}}_k(K,K)$, whose dimension over $K$ is $[K:k]$. Hence $m\le [K:k]$. In particular, a finite automorphism group $G$ satisfies $|G|\le [K:K^G]$.