Paper 3, Section II, D

Let $G$ be a finite group, $X$ the set of proper subgroups of $G$. Show that conjugation defines an action of $G$ on $X$.

Let $B$ be a proper subgroup of $G$. Show that the orbit of $G$ on $X$ containing $B$ has size at most the index $|G: B|$. Show that there exists a $g \in G$ which is not conjugate to an element of $B$.

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