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Find all cyclic subgroups of $\mathbb{Z}_{24}$.

I know the subgroups of $\mathbb{Z}_{24}$ to be $\mathbb{Z}_{24}, 2\mathbb{Z}_{24}, 3\mathbb{Z}_{24}, 4\mathbb{Z}_{24}, 6\mathbb{Z}_{24}, 8\mathbb{Z}_{24}, 12\mathbb{Z}_{24}, 24\mathbb{Z}_{24}=\langle0\rangle$.

Which of these are cyclic and why?

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1 Answer 1

up vote 3 down vote accepted

As $\mathbb{Z}_{24}$ is cyclic (it is generated by the element $1$), and a subgroup of a cyclic subgroup is cyclic, all the subgroups are cyclic.

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Why is $\mathbb{Z}_{24}$ cyclic? –  Desperate Fluffy Sep 16 '13 at 3:30
    
Well $\mathbb Z_{24}$ is assumed to be $\mathbb Z/24\mathbb Z$. Do you mean it not to be? –  Ian Coley Sep 16 '13 at 3:31
    
$\mathbb{Z}_{24} = \langle 1\rangle$. –  Michael Albanese Sep 16 '13 at 3:31
    
@IanColey No, just wondered. Its cyclic groups that I'm having troubles with. –  Desperate Fluffy Sep 16 '13 at 4:02

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