Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In an additive number group (e.g. $(\mathbb{Z},+)$) there is a well known notation for absolute value, namely $|a|$, which coincides with $\max(a,-a)$, for $a \in \mathbb{Z}$.

When the context is a multiplicative number group instead, is there a similar notation, which would coincide with $\max(a,\frac{1}{a})$?

share|cite|improve this question
Well known? In general, we don't have an ordering on an additive (abelian) group $G$, and you need an ordering on your group to even say $\max(a,b)$. Otherwise, what does $\max$ even mean? (For example, what is $\left|-3\right|$, where $-3\in\Bbb{Z}/5\Bbb{Z}$?) Do you mean a subgroup of $\Bbb{R}$? – Stahl Apr 5 '13 at 20:57
There is no such absolute value in an arbitrary additive group. Where did you get that idea? – Thomas Andrews Apr 5 '13 at 20:58
I forgot to emphasize that I mean number groups. I will change the question correspondingly. – Chiel ten Brinke Apr 5 '13 at 21:01
Also, I'd note that $\left|a\right|$ is very often used to mean the order of the subgroup of $G$ generated by $a$ (which is defined to be the order of the element $a$): $\left|a\right| := \left|\left<a\right>\right|$. – Stahl Apr 5 '13 at 21:02
If you talk ab out a subgroup of $\mathbb R>0$, then $\max\{a,\frac1a\}$ is fine - but is not really something different from the standard absolute value on the isomorphic additive group $\mathbb R$. Using additive or multiplicative notation is arbitrary. – Hagen von Eitzen Apr 5 '13 at 21:07
up vote 2 down vote accepted

If you're working with a multiplicative group $G\subseteq\Bbb{R}$, you can definitely say $$ \operatorname{abs}(g) := \max\{g,g^{-1}\}\quad\textrm{for }g\in G. $$ The question is whether or not it is useful to the study of the group $G$ in any way.

Also, when it comes to the question of notation, $\left|g\right|$ is normally used to mean the order of $g\in G$, which is the smallest $n\in\Bbb{N}$ such that $g^n = e$ (where $e\in G$ is the identity) or equivalently, the order of the subgroup of $G$ generated by $g$. As far as I know, there is no standard notation for $\max\{g,g^{-1}\}$ when $g\in G\subseteq\Bbb{R}$.

share|cite|improve this answer

What is $\max$ in arbitrary group? It is used only for ordered groups (independently, is the group additive or multiplicative).

share|cite|improve this answer
I think this is a comment, if I may. – 1015 Apr 5 '13 at 21:00
I disagree: I think it is a good answer to the question as it currently stands. – Pete L. Clark Apr 5 '13 at 21:03
@julien: Which is your answer? – Boris Novikov Apr 5 '13 at 21:06
Ok, I did not may. Sorry, then. – 1015 Apr 5 '13 at 21:22

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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