Is primitive root or order in number theory relate with order of element in cyclic group?

Question

I just read abstract algebra and found notation of cyclic group (I don't read the whole yet ) the order in number theory state $ a^{b}\equiv 1(mod N)$ and Cyclic group state $ a^{n}=e$ or I not sure it can't write as $a^{n}\equiv 0(mod n)$ Both consider by congruence. Is it has relationship between them?

Answer 1

It's useful to use additive notation for the cyclic group $\Bbb Z/n$ in this case it's $ka\equiv 0\mod n$ is the additive order, where the group operation is addition. The $a^k\equiv 1\mod n$ is acutally a question in the group $\Bbb Z/n^*$ of elements of $\Bbb Z/n$ which are coprime to $n$, this is a group under multiplication and when they say "order" in that sense they mean multiplicative order. So really these things are happening in separate groups, they're just abusing notation.

As a side note: "order" is a general notion for a group: it's not just for cyclic groups.