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Let $G$ be a group of order $n\cdot p^m$. Let $K$ be the set of all subsets of $G$ of size $p^m$, and let $G_j$ be the group stabilizing $j$ in $K$. Why is $\rm{ord}(G_j) \leq |j|$?

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Fix any $a \in j$ consider the map $\varphi:G_j \to j$ defined by $\varphi(g)=ga$. This map is an injection since if $g_1a=g_2a$, then you can multiply by $a^{-1}$ from the right and get $g_1=g_2$. Hence $|G_j| \leq |j|$.

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