If G has no nontrivial subgroups, prove that G must be finite of prime order.

Expert Solution

Colby Messinger answered 2 years ago

If all proper nontrivial subgroups of a nontrivial group are isomorphic to each other, must G...

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(a) Show that a group that has only a finite number of subgroups must be a finite group. (b) Let G be a group that has exactly one nontrivial, proper subgroup. Show that G must be isomorphic to Zp2 for some prime number p. (Hint: use part (a) to conclude that G is finite. Let H be the one nontrivial, proper subgroup of G. Start by showing that G and hence H must be cyclic.)

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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...

Question

If G has no nontrivial subgroups, prove that G must be finite of prime order.

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