= Normalizer condition for finite p-groups
{title2=$H<G\ \Longrightarrow\ H<N_G(H)$}
Every proper <subgroup> $H$ of a <finite p-group> is properly contained in its <normalizer>. Let $H$ act on the left <cosets> $G/H$ by left multiplication. The fixed cosets are those represented by elements of $N_G(H)$, so there are $[N_G(H):H]$ of them. All other <orbits of a group action> have sizes divisible by $p$. Since $[G:H]$ is divisible by $p$ and at least one coset is fixed, the fixed-coset count is a positive multiple of $p$. This also proves $[N_G(H):H]\geq p$.
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