(1) Suppose X is a G-set. For each x € X, let Gx := {ge G:g x= x } denote the stabilizer of x. For every subgroup H of G, let N(H) := {g €G:gHg¹ = H} XH := { x = X : hx = x for every h E H} (a) (b) Show that N(H) is a subgroup of G. Show that XH is naturally a N(H)-set, i.e. n · x € XH for every x € XH and n = N(H). (c) Suppose G is a finite group act transitively on X and x € X. Show that N(G) acts transitively on XG. (Recall that a group G acts transitively on X if and only if there is a g E G such that gx = y for every y E X.)

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