# Quick subgroup question

• October 13th 2011, 11:10 AM
tangibleLime
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)
• October 13th 2011, 12:55 PM
Deveno
Re: Quick subgroup question
what do you mean, "sets generated by <n>"?
• October 13th 2011, 12:58 PM
tangibleLime
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)
• October 13th 2011, 02:06 PM
Deveno
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.