My problem is how to find all groups which have one exactly non-proper subgroup. Thanks
Finding all groups with given property
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0Looking 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. ā 2012-06-06
3 Answers
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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5All groups have only one non-proper subgroup... The group itself. ā 2012-06-06
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 $nā„2$. Now think of the possibilities of $n$ (Hint: apply sylow first Theorem).
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.