(a) Show that a group that has only a finite number of subgroups must be a...

(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.)

Expert Solution

Colby Messinger answered 1 year ago

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

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Question

(a) Show that a group that has only a finite number of subgroups must be a...

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