TITLE: What's the smallest difference in a group?
QUESTION [5 upvotes]: In the additive group $\Bbb Z$ we can fairly unambiguously say $1$ is the smallest difference between two elements.
I guess a more rigorous statement might be to give $\Bbb Z$ its topology as a subspace of $\Bbb R$ and then to say that the pair $\{1,-1\}$ are the set that share the smallest absolute value: $\{1,-1\}=\{x\in \Bbb Z:\lvert x\rvert=\min\{\lvert x\rvert:x\in \Bbb Z\}\}$.
Is there a general form for the smallest interval in a group?  Does it equal the smallest nonzero element in general?

REPLY [2 votes]: This answer mainly comes from Dale Rolfsen’s Ordered Groups and Topology.

Amazingly, sometimes groups do have smallest elements!
Before we ask questions about the smallest element, we should define what we mean. 

A left-ordering on a group $G$ is a total order $<$ on the group so that if $g<h$ then $xg < xh$ for every $x\in G$. 

Not all groups are left-orderable! Notice that the invariance under multiplication shows that any left-orderable group has no finite order elements. (Prove this yourself!). Given the identity element $e\in G$, let’s define the smallest element of $G$ to be the smallest element of the set 
$$\mathcal{P}^+ = \{g\in G ~|~ e<g\}.$$
This is called the positive cone of $G$. Some examples of left-orderable groups are:

Free groups
Surface groups
Mapping class groups of surfaces with boundary
In particular, braid groups

and all their subgroups. Now, a sufficient condition to have a smallest element is for the order $<$ to be a well order. 

An order $<$ on $G$ is a well order if any subset $S$ of $G$ has an element $s\in S$ such that $s<x$ for all other $x\in S$.

This property, applied to the positive cone $\mathcal{P}^+$, allows us to prove that $G$ would have a smallest element.

Fact: The positive pure braid group is well ordered under the Magnus ordering. Thus, it has a smallest element. This is also true for the positive braid group under the Dehornoy ordering.

However, the full (pure) braid group itself under this order is order dense; that is, for any $g<h$ there is some $x$ so that $g<x<h$. Thus, it can never have a smallest element.
It isn’t clear to me if there are algebraic obstructions to having a well ordering, what the smallest element is in the two above examples, or whether you can have a smallest element without having a well order.