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?

  • 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

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 $\bar x$?

  • $\begingroup$ 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

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