Talk:Cyclicity Condition for Units of Ring of Integers Modulo n

Added explanation

Maybe we should cite a theorem that direct product of two groups contains them as a subgroup?

I read from Wikipedia that

the isomorphic copy of G in G × H is the product G × {1} , where {1} is the trivial subgroup of H.

Does $\mathsf{Pr} \infty \mathsf{fWiki}$ have this theorem? If not we can create a page for it. --Hbghlyj (talk) 13:19, 17 March 2024 (UTC)

Add a generalization to Group Direct Product of Cyclic Groups?

We can add a generalization to the theorem Group Direct Product of Cyclic Groups that if a cyclic group is the direct product of $n$ groups then all are cyclic.

A product $G=C_{n_1} \times \cdots \times C_{n_r}$ is not cyclic unless $n_1, \ldots, n_r$ are coprime.

The direction we need is $\Leftarrow$.

Proof of $\Leftarrow$: every element in this group has order dividing the lowest common multiple $\left[n_1, \ldots, n_r\right]$, which is strictly smaller than $\# G$.

Then we can cite the theorem here. --Hbghlyj (talk) 13:34, 17 March 2024 (UTC)

Definition of Primitive root modulo $n$

According to MathWorld and Wikipedia, the generator of $U = \struct {\paren {\Z / n \Z}^\times, \times}$ is called Definition:Primitive Root Modulo n. I didn't find this definition page on ProofWiki.

--Hbghlyj (talk) 10:31, 2 May 2024 (UTC)

Using Wikipedia as a source is heavily non-recommended, for reasons which are well-documented on this site which I would rather not go into all over again.

It is also recommended that Mathworld similarly be mistrusted, unless such material can be sourced from a hard copy. --prime mover (talk) 10:49, 2 May 2024 (UTC)

Also in This article incorporates material from Primitive root on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

--Hbghlyj (talk) 10:56, 2 May 2024 (UTC)