# Notation for “absolute value” in multiplicative group.

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 $\max(a,\frac{1}{a})$?

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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

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 $g\in G\subseteq\Bbb{R}$.

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

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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