Algebra and Geometry | Part IA, 2005

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

Let GG be a finite group and XX the set of its subgroups. Show that g(K)=gKg1g(K)=g K g^{-1} (gG,KX)(g \in G, K \in X) defines an action of GG on XX. If HH is a subgroup of GG, show that the orbit of HH has at most G/H|G| /|H| elements.

Suppose HH is a subgroup of GG and HGH \neq G. Show that there is an element of GG which does not belong to any subgroup of the form gHg1g H g^{-1} for gGg \in G.

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