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Is there a special term for groups, the only subgroups of which are the trivial group and the group itself? Obviously, any such group is a trivial group or a cyclic group of prime order.

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And in fact, "cyclic group of prime order" is the usual name for them. – Arturo Magidin Mar 15 '12 at 16:13

Yes. They're abelian simple groups, and yes, you've found them all.

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