Let G be a group of order 42 = 2 * 3 * 7 (a) Let...

Let G be a group of order 42 = 2 * 3 * 7

(a) Let P7 be a Sylow 7-subgroup of G and let P3 be a Sylow 3-subgroup of G . What are the orders of P3 and P7?

(b) Prove that P7 is the unique Sylow 7-subgroup of G and that P7 is normal.

(c) Prove that P3P7 is a subgroup of G

(d) Prove that P3P7 is a normal subgroup of G .

(e) Let P2 be a Sylow 2-subgroup of G . Prove that G \cong (P3P7) \Join P2

(f) Assume subgroup not abelian. WHat is the index of N G(P3) in G ? [G: N G(P3) ] = _______

Expert Solution

Colby Messinger answered 1 year ago

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