# Thread: multiplicative groups cyclic and isomorphic

1. ## multiplicative groups cyclic and isomorphic

How do I determine if a multiplicative group, say $\mathbb{Z}_{10}^{*}$ is cyclic.

Also how would I determine if two multiplicative groups, say $\mathbb{Z}_{7}^{*},\mathbb{Z}_{14}^{*}$, are isomorphic or not.

2. Originally Posted by magus
How do I determine if a multiplicative group, say $\mathbb{Z}_{10}^{*}$ is cyclic.

Also how would I determine if two multiplicative groups, say $\mathbb{Z}_{7}^{*},\mathbb{Z}_{14}^{*}$, are isomorphic or not.

Read here Multiplicative group of integers modulo n - Wikipedia, the free encyclopedia

Tonio

3. I've looked there. Unfortunately they talk about homomorphisms and I'm looking for isomorphisms. I looked at the linked page on mathworld and saw the cycle graphs and can see how multiplicative groups with the same cycle graph being isomorphic.

Do you construct these cycle graphs to determine if two multiplicative groups are isomorphic? To determine if it's cyclic do you just have to try trial and error until you find a generator (or don't find one)?

Also I found this

" $M_m$ is a cyclic group (which occurs exactly when m has a primitive root) iff m is of one of the forms $m=2,4,p^2,2p^2$, where p is an odd prime and (Shanks 1993, p. 92). The first few of these are m=3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, ... (Sloane's A033948; Shanks 1993, p. 84)."

from mathworld. I'd like to use it but I don't think I should without proof. The problem is that I'm not even sure where to start proving it.

I guess in short: I read it and nothing is jumping out at me.