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My problem is how to find all groups which have one exactly non-proper subgroup. Thanks

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It cannot be infinte. – Babak S. Jun 6 '12 at 13:23
Could it be you actually meant "...which have exactly one non-trivial proper subgroup"? – DonAntonio Jun 6 '12 at 13:40
Looking at the answers below, it is clear that some posters are counting the one-element subgroup, and others are not. The question as stated asks for answers that count the one-element subgroup, but mathematicians tend to disregard it (hence DonAntonio's comment). It wouldn't hurt to be extra explicit about what you want here. – Brett Frankel Jun 6 '12 at 14:20
up vote 4 down vote accepted

Groups with only one proper subgroup

A nontrivial group $G$ has no proper subgroups except the trivial group iff $G$ is finite and of prime order.

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All groups have only one non-proper subgroup... The group itself. – Seth Jun 6 '12 at 13:30

Obviously and as Matt noted, your group cannot be infinte. Moreover there are not two distinct prime numbers $p$ and $q$ which divide the order of $G$. (Why?). So the order of group is $p^n$ for some $n≥2$. Now think of the possibilities of $n$ (Hint: apply sylow first Theorem).

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Assuming you mean groups with exactly one proper subgroup:

Take a group $G$ and an element $g\in G$ and generate a cyclic subgroup. There are two cases: either every element $g\in G$ generates a proper subgroup, or there is an element $g \in G$ that generates the whole group. In both of these cases you can analyze the possibilities.

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