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    Exclamation nontrivial subgroup

    Show that a group G has no nontrivial subgroups if and only if it is a cyclic group of prime order
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by snick View Post
    Show that a group G has no nontrivial subgroups if and only if it is a cyclic group of prime order
    If G=\{e\} we're done. So, assume not. Then, let g\in G-\{e\} we must clearly have that \langle g\rangle\leqslant G but it isn't trivial and so it must be improper. Thus, G=\langle g\rangle. Now, if m\mid |G| then by G's cyclicness we must have that there exists some H\leqslant G such that |H|=m. Since H cannot be nontrivial or proper it follows that m=1,|G|. The conclusion follows.
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