The question is: Let G be a finite group, H, K be normal subgroups of G,...

The question is: Let G be a finite group, H, K be normal subgroups of G, and H∩K is also a normal subgroup of G. Using Homomorphism theorem ( or First Isomorphism theorem) prove that G/(H∩K) is isomorphism to a subgroup of (G/H)×(G/K). And give a example of group G with normal subgroups H and K such that G/(H∩K) ≆ (G/H)×(G/K), with explanation.

I was trying to find some solutions for the isomorphism proof part, but they all seems to have the condition with H∩K = {e} . I can ensure that there is no missing condition in my question. As there is another subquestion which I've already know the solution, is about given H∩K = {e} and show G/(H∩K) is isomorphism to (G/H)×(G/K).

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Expert Solution

Colby Messinger answered 1 year ago

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