Other proofs that subgroups of $A_5$ have order at most 12

abstract-algebra group-theory finite-groups permutations

Let $H$ be a proper subgroup of $A_5$ with $|H| > 12$, and let $A_5$ act on the set of left cosets of $H$ by left multiplication. Then, you should be able to see that the kernel of this action must be strictly between $1$ and $A_5$.

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