Let G be a group and let N and K be normal subgroups of G. (a) Prove that for all a e N and be K, aba-b-1 € Nn K. Hint: Recall that if T is a normal subgroup of G, then for all a e G, xTx- cT. (b) Suppose N nK = {ec}. Prove that for all a e N and be K, ab = ba. (c) Suppose N n K = {ec}. Prove that f:N x K → G, f(a, b) = ab is an injective homomorphism. (d) Suppose G is a finite group, G| = |N||K], and Nn K = {ec}. Prove that f is an isomorphism. (e) Suppose |N| = 4, |K| = 9, and |G| = 36. Prove that G is abelian and determine the different possible elementary divisors and the different possible invariant factors. Use these to determine the possible groups that G is isomorphic to in the Funda- mental Theorem of Finitely Generated Abelian Groups, in the form of elementary divisors and invariant factors. Hint: Note to use the previous part of the question, you must prove that Nn K = {eg}. %3D

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