3.II.5D

Define the notion of an action of a group $G$ on a set $X$. Assuming that $G$ is finite, state and prove the Orbit-Stabilizer Theorem.

Let $G$ be a finite group and $X$ the set of its subgroups. Show that $g(K)=g K g^{-1}$ $(g \in G, K \in X)$ defines an action of $G$ on $X$. If $H$ is a subgroup of $G$, show that the orbit of $H$ has at most $|G| /|H|$ elements.

Suppose $H$ is a subgroup of $G$ and $H \neq G$. Show that there is an element of $G$ which does not belong to any subgroup of the form $g H g^{-1}$ for $g \in G$.

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