Are the maximal subgroups of a finite group always conjugate? Suppose that $H$ is a maximal subgroup of $G$. Consider a subgroup $H$ conjugate to any element of $G$, this subgroup has the same order as $H$, hence max is maximal and all maximal subgroups are conjugate. Is this reasoning correct?
2026-03-26 13:30:24.1774531824
Maximal subgroups of finite group
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All conjugates of maximal subgroups are maximal, but not all maximal subgroups are conjugate of each other. They don't even need to be the same order: take a product of two cyclic groups of orders two different primes.