For a group $G$, let $\mathcal F(G)$ denote the collection of all subgroups of G. Which of following can occur?

A) $G$ is finite, but $\mathcal F(G)$ is infinite

B) $G$ is infinite but $\mathcal F(G)$ is finite

C) $G$ is countable but $\mathcal F(G)$ is uncountable

D) $G$ is uncountable but $\mathcal F(G)$ is countable

option A,B is false, O think option D is also false, but how? Can someone give me a hint?

  • 1
    $\begingroup$ If $G$ is uncountabe then there must be uncountably many cyclic subgroups generated by single elements. $\endgroup$ – Derek Holt Dec 1 '15 at 21:33

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