# Units group of Z modulo

1. Nov 7, 2011

### Bachelier

I came accross this question. Is the group (ℤ/34/ℤ)x cyclic?
We haven't discussed the theorem in class that any units group of Z modulo n is iff n = 1, 2, 4, pk and 2pk (where p is an odd prime). But thanks to Deveno I know about it. So in this case, p =17 works, so the group is cyclic?

but is there a different way to show it besides using brute force .

2. Nov 7, 2011

### Bachelier

Re: (ℤ/34/ℤ)^x

BTW, p doesn't have to be equal to 4k+3 for this to be true, correct?

Last edited: Nov 7, 2011
3. Nov 8, 2011

### Bachelier

Re: (ℤ/34/ℤ)^x

I guess I can show that ℤ/34/ℤx is ≈ to ℤ/17/ℤx and by the field's finite subgroup theorem, it is a cyclic.