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Is there a simple characterization for a group $G$ satisfying $$(\forall H,K\le G)(H\subseteq K\text{ or } K\subseteq H)$$

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up vote 6 down vote accepted

Hint: First, Assume that $x\in G$ with infinite order. Compare $x^p$ and $x^q$ where $p,q$ are distinct prime numbers. Now Suppose that $G$ is torsion. Compare elements with order $p$ and $q$ where $p,q$ are distinct prime numbers.

In next step. you should show that $G$ is abelian group.

Finally, you should got $G=\mathbb Z_{p^n}$ where $n\in \mathbb N\cup \{\infty\}$

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well if $G$ is not torsion-free it is a p-group. what if it is torsion-free? –  user795571 Aug 16 at 13:52
    
@user138171 $G$ cannot contain an element with infinite order. –  Babak Miraftab Aug 16 at 14:02

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