terhorst
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Problem
Suppose for all subgroups H,K of a finite group G, either H \subset K or K \subset H. Show that G is cyclic and its order is the power of a prime.
Attempt
I think I get the intuition: if H and K are not the same, then one of them must be the trivial subgroup and the other must be G itself. So if g \in G but g \notin H, then \left\langle g \right \rangle is a subgroup containing g, so by hypothesis, H \subset \left\langle g \right \rangle. From here I want to show that H is actually the trivial subgroup. No idea yet about the power of a prime thing. Can anyone provide a hint? Thanks!
Suppose for all subgroups H,K of a finite group G, either H \subset K or K \subset H. Show that G is cyclic and its order is the power of a prime.
Attempt
I think I get the intuition: if H and K are not the same, then one of them must be the trivial subgroup and the other must be G itself. So if g \in G but g \notin H, then \left\langle g \right \rangle is a subgroup containing g, so by hypothesis, H \subset \left\langle g \right \rangle. From here I want to show that H is actually the trivial subgroup. No idea yet about the power of a prime thing. Can anyone provide a hint? Thanks!