Is the action by conjugation of $S_5$ on its six $5$-Sylow subgroups faithful?
I can see how the second Sylow theorem makes this action transitive.
I am not sure how to check whether it's faithful, without actually doing the computations.
2026-04-12 11:36:11.1775993771
Is the action by conjugation of $S_5$ on its six $5$-Sylow subgroups faithful?
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$A_5$ is simple and cannot be the kernel.