# A observation on conjugate subgroups

Originally published at 狗和留美者不得入内. You can comment here or there.

Let $H$ and $H'$ be conjugate subgroups of $G$, that is, for some $g \in G$, $g^{-1}Hg = H'$. Equivalently, $HgH' = gH'$, which means there is some element of $G/H'$ such that under the action of $H$ on $G/H'$, its stabilizer subgroup is $H$, all of the group of the group action. Suppose $H$ is a $p$-group with index with respect to $G$ non-divisible by $p$. Then such a fully stabilized coset must exist by the following lemma.

If $H$ is a $p$-group that acts on $\Omega$, then $|\Omega| = |\Omega_0|\;(\mathrm{mod\;} p)$, where $\Omega_0$ is the subset of $\Omega$ of elements fully stabilized by $H$.

Its proof rests on the use orbit stabilizer theorem to vanish out orbits that are multiples of $p$.

This is the natural origin of the second Sylow theorem.

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