Order of Elements in a Group: A Quick Check of Understanding

[Artusartos]

I just want to check if there is anything wrong with my understanding...

Let's say we have a group of order 42 that contains $Z_6$. Since the group of units of $Z_6$ has order (3-1)(2-1), it means that we have 2 elements of order 6 in G, right? In other words, for any cyclic subgroup of order n, we just calculate the group of units to see how many elements of order n we have. Is that correct? Also, if we let $P=Z_6$, we can have at most 7 cyclic subgroups of order 6, since 7=42/|N(P)| (where N(P) is the normalizer of P), if N(P)=6 (which is the smallest it can be, since it has to contain $Z_6$). I'm just wondering if my understanding is correct.

Thanks in advance

 

[fresh_42]

Not right. We might have elements of order $6$ in $G$ or not, but your reasoning with the units of $\mathbb{Z}_6$ is strange. The non-units you might refer to are $0$ and $3$ and none of which has order $6$.