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Suppose for all subgroups [tex]H,K[/tex] of a finite group [tex]G[/tex], either [tex]H \subset K[/tex] or [tex]K \subset H[/tex]. Show that [tex]G[/tex] is cyclic and its order is the power of a prime.

Attempt

I think I get the intuition: if [tex]H[/tex] and [tex]K[/tex] are not the same, then one of them must be the trivial subgroup and the other must be [tex]G[/tex] itself. So if [tex]g \in G[/tex] but [tex]g \notin H[/tex], then [tex]\left\langle g \right \rangle[/tex] is a subgroup containing [tex]g[/tex], so by hypothesis, [tex]H \subset \left\langle g \right \rangle[/tex]. From here I want to show that [tex]H[/tex] is actually the trivial subgroup. No idea yet about the power of a prime thing. Can anyone provide a hint? Thanks!

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# Homework Help: Subgroups of a p-group

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