# Show that there exists a unique subgroup of $Z_n$ of order $d$ generated by $[k].$

Suppose that $$d$$ and $$k$$ are positive integers satisfying $$dk = n$$. Show that there exists a unique subgroup of $$Z_n$$ of order $$d$$ generated by $$[k].$$

Then, suppose that the cyclic group $$G$$ operates on a set $$S$$ and $$g_1$$ and $$g_2$$ generate $$G$$. Show that #fixed $$g_1$$ = # fixed $$g_2$$.

I first showed that the number of generators of $$Z_n$$ was $$\phi(n)$$ but I don't know how to proceed. Any help would be great, thank you in advance!

• Well, first show that $\langle k\rangle$ has order $d$, then show that any subgroup of order $d$ must contain $k$. – Don Thousand Mar 26 at 21:14
• Yes, @DonThousand, that was my thought too but I'm not sure how to go about either of those steps. – James Done Mar 26 at 21:17
• The first part is pretty easy ... Think about multiples of $k$. The second part, note that subgroups of cyclic groups are cyclic... – Don Thousand Mar 26 at 21:19
• Ok, I think I got the first part now. But I still don't understand how to do the "must contain k" part of it. Sorry, thank you for your time! – James Done Mar 26 at 21:23
• Note that any subgroup of order $d$ must contain an element of order $d$. As such, it must be $g^k$ for some generator $g$. What should that tell you? – Don Thousand Mar 26 at 21:24

I first showed that the number of generators of $$\mathbb{Z}/n$$ was $$\phi(n)$$ ...
For the first part: since the multiples of $$k$$ are a subgroup of $$\mathbb{Z}$$, their reductions mod $$n$$ are a subgroup of the integers mod $$n$$. How many different multiples of $$k$$ are there mod $$n$$?
For the second part: since $$g_1$$ generates the group, $$g_2=g_1^m$$ is some power of $$g_1$$. Now, if $$x$$ is a fixed point for the action of $$g_1$$, show that it's also a fixed point for the action of $$g_2$$.
Then, switch. Since $$g_2$$ generates the group...
• Ok, this makes sense but how would I show that if x is a fixed point for the action of $g_1$, it is also a fixed point action for the action of $g_2$ (my textbook doesn't explain fixed points at all). Thank you @jmerry! – James Done Mar 26 at 21:54