Are all sets generated by each <n> from $\displaystyle \mathbb{Z}_n$ subgroups? (I know that they are not all generators, and some subgroups contain others)

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- Oct 13th 2011, 11:10 AMtangibleLimeQuick subgroup question
Are all sets generated by each <n> from $\displaystyle \mathbb{Z}_n$ subgroups? (I know that they are not all generators, and some subgroups contain others)

- Oct 13th 2011, 12:55 PMDevenoRe: Quick subgroup question
what do you mean, "sets generated by <n>"?

- Oct 13th 2011, 12:58 PMtangibleLimeRe: Quick subgroup question
For example, if $\displaystyle \mathbb{Z}_{5}$,

<1>, <2>, <3>, <4>

(which are all $\displaystyle \mathbb{Z}_{5}$, so maybe that's a bad example, but I think it gets my intention across) - Oct 13th 2011, 02:06 PMDevenoRe: Quick subgroup question
are you asking if <k> = $\displaystyle \mathbb{Z}_n$ for every k in $\displaystyle \mathbb{Z}_n$, or are you asking if every subgroup of $\displaystyle \mathbb{Z}_n$ is cyclic (and thus equal to <k> for some k)?

the answer to the first is no, the answer to the second is yes.