What's the smallest difference in a group? 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?

In general, groups that can be endowed with an order that is compatible with the binary operation - that is, left- and/or right-invariant, in the sense that $$\forall a,b,c,\qquad a<b\implies ca<cb\quad(\text{respectively, }ac<bc)$$  are very sparse in the landscape of groups. Apart from $\Bbb Z$, all subgroups of $\Bbb R$ can be - and you can already see that in $\Bbb Q$ and $\Bbb R$ there is no smallest difference (or smallest element).
The braid group is one quite non-trivial example where this can be done, although the Dehornoy order is only left-invariant (no order on the braid group $B_n,n\geq 3$ can be both left- and right-invariant).
In the rare cases where there is such an order, then you can define positive to mean larger than the identity element, and it is immediate that the "smallest difference", if any, is going to be equivalent to the smallest element of the group in the sense that any difference between two elements can be translated into the difference between some element and the identity. 
In the group $\Bbb S^1$ for instance, you can see that there is no smallest difference, and that it is impossible to define a compatible order. In the group $\Bbb Z^2$, you could define the smallest difference by using the distance $\sqrt{x^2+y^2}$; this would be compatible with the binary operation (translations don't distort distances), but not with any order.
