# Units in groups

This might seem trivial, but can I call an element of a group (with multiplication as its induced operation) a unit? I'm asking this because Fraleigh (from his book A First Course in Abstract Algebra, p. 187) referred to this theorem

The set $G_{n}$ of nonzero elements of $\mathbb{Z}_{n}$ that are not 0 divisors forms a group under multiplication modulo $n$.

to claim that $a\in\mathbb{Z}_{n}$ is a unit in this theorem

Let $m$ be a positive integer and let $a\in\mathbb{Z}_{m}$ be relatively prime to $m$. For each $b\in\mathbb{Z}_{m}$, the equation $ax=b$ has a unique solution in $Z_{m}$.

His proof is as follows

By Theorem 20.6, $a$ is a unit in $\mathbb{Z}_{m}$ and $s=a^{-1}b$ is certainly a solution of the equation. Multiplying both sides of $ax=b$ on the left by $a^{-1}$, we see that this is the only solution.

Theorem 20.6 is the first theorem I presented by the way.

As far as I know, the term "unit" is only used in Ring Theory, but given the definition, I can see why he would use such a term for a group.

I understand that an element being a member of a group with multiplication as its induced operation automatically means its multiplicative inverse exists. I just don't understand why he would use the term "unit".

I've tried searching for answers all over the internet, but none seems to point out that the terminology is valid. Even the Wikipedia page about groups does not contain the word "unit".

• In a group, every element has inverse w.r.t the group operation. Are you sure Fraleigh isn't treating $\mathbb{Z}_n$ as a ring with sum and multipilcation? Could you link the paper or book where he claims that $a\in\mathbb{Z}_n$ is a unit?
– Javi
Feb 28, 2018 at 11:23
• Wait, I'll edit. Feb 28, 2018 at 11:25
• Okay, I think I now know the answer. Correct me if I'm wrong, but I think the group $G_{n}$ was just used to show that all such elements of $\mathbb{Z}_{n}$ with the mentioned properties are units in $\mathbb{Z}_{n}$. Feb 28, 2018 at 12:00
• Yeah, that's it. And he's talking about ring units.
– Javi
Feb 28, 2018 at 12:08

($\mathbb{Z}_n,+,\cdot)$ is a ring.