Throughout this question, let G be a finite group, let p be aprime, and suppose that H ? G is such that [G : H] = p.
Let G act on the set of left cosets of H in G by leftmultiplication (i.e., g · aH = (ga)H). Let K be the set of elementsof G that fix every coset under this action; that is,
K = {g ? G : (?a ? G) g · aH = aH}.
(a) Prove that K is normal subgroup of G and K?H. From theresult of part (a) it follows that K ? H. For the remainder of thisproblem, we let k = [H : K].
(b) Prove that G/K is isomorphic to a subgroup ofSp.
(c) Prove that pk | p!. Hint: Calculate [G : K].
(d) Now suppose, in addition to the setup above, that p is thesmallest prime dividing |G|. Prove that H is normal subgroup of G.Hint: Show that k = 1.
