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