0
$\begingroup$

My university's textbook on Abstract Algebra states:

Let $n\in \mathbb{N}_0$. We note the group of units of the quotient ring $\langle\mathbb{Z}_n,+,\cdot\rangle$ as $\langle\mathbb{Z}^\times_n,+,\cdot\rangle$, where $\mathbb{Z}^\times_n = \{\bar x\mid x\in \mathbb{Z}\;\wedge\; \gcd(x,n)=1\}$.

Given that $\gcd(x,n)=1$ is equivalent to $x$ being invertible in $\langle\mathbb{Z}_n,\cdot\rangle$, it's easy to prove that such $x$ form a multiplicative group (for any $x_1$ and $x_2$, $x_1x_2$ is invertible as well with inverse $x_2^{-1}x_1^{-1}$ and for any $x$, $x^{-1}$ is also invertible with inverse $x$).

However, I am doubting the textbook's use of $+$ in the context of $\mathbb{Z}^\times_n$ being a group. Of course, addition is still a valid algebraic operator, but I don't think it's closed in $\mathbb{Z}^\times_n$, letalone a group operator. For example, $1$ is always in $\mathbb{Z}^\times_n$, but $1+1$ is not.

Am I missing something here, or is their notation/phrasing sloppy?

$\endgroup$
1
  • 1
    $\begingroup$ Probably a copy-paste-modify error, they forgot to remove the "$+$". For every unit $u$, $-u$ is also a unit, and $0 = u + (-u)$ isn't. $\endgroup$ May 21 '20 at 15:55
2
$\begingroup$

No it's definitely not a group because the identity $0$ is not there.

It should read $\langle {\mathbb Z^\times_n, \cdot}\rangle$, not $\langle {\mathbb Z^\times_n, +, \cdot}\rangle$ which is the symbolism for a ring, not a group.

Which textbook is this?

Incidentally, what is meant by $\bar x$?

$\endgroup$
2
  • $\begingroup$ Thanks for clarifying. This is our university's homebrew textbook (ISBN: 90-5350-771-X). $\bar x$ is the residue class with representative $x$. You made me wonder, though: is there certain notation to indicate that a ring $G$ is a group w.r.t. both its multiplication (not counting 0, then) and addition (so really, notation that it's a field)? $\endgroup$
    – Mew
    May 21 '20 at 16:10
  • $\begingroup$ Not as far as I know -- that is there's no notation to distinguish $\langle{S, +, \cdot}\rangle$ being a ring from it being a field. What you'd do is state whether $\langle{S^*, \cdot}\rangle$ was a semigroup or a group. $\endgroup$ May 21 '20 at 16:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.