# Math Help - Quick subgroup question

1. ## Quick subgroup question

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

2. ## Re: Quick subgroup question

what do you mean, "sets generated by <n>"?

3. ## Re: Quick subgroup question

For example, if $\mathbb{Z}_{5}$,

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

(which are all $\mathbb{Z}_{5}$, so maybe that's a bad example, but I think it gets my intention across)

4. ## Re: Quick subgroup question

are you asking if <k> = $\mathbb{Z}_n$ for every k in $\mathbb{Z}_n$, or are you asking if every subgroup of $\mathbb{Z}_n$ is cyclic (and thus equal to <k> for some k)?

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