2. Let G be a group of order #G = p'm where r > 0 is an integer, p is a prime number, and p does not divide m. Let S C G be a p-Sylow subgroup of G and let X = G/S is the set of left cosets. (a) Show that p does not divide #X. (b) Let H be a subgroup of G. Let H act on X by h.gS = (hg)S for h = H, gS E X. i. Show that if h E H is such that the left cosets hS and hgs are equal, then there is s ES such that h = gsg-¹ 1 ii. Deduce that the stabilizer Has = gSg-¹nH.

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A: The given equation is

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A:

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A:

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A: Solution:

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A: Given information:The given equation is