On normal forms of group elements

Question

I was reading about finitely presented groups. Some finitely presented groups (like the braid groups) have normal forms, which means that every element of that group has a unique representation of sorts. I wanted a few examples of finitely presented groups without normal forms. I was wondering if we have a finitely presented group, can we prove that the given group does not have a normal form?

Answer 1

You can define a normal form for any finitely generated group.

Suppose $G$ is generated by the finite set $X$ and let $A = X \cup X^{-1}$. As your representative of the group element $g$, choose the least word in the alphabet $A$ that represents $g$ under some well-ordering of the set of words in $A$.

An example of such a well-ordering is a lenlex ordering. For that, you choose some arbitrary total ordering $<$ of $A$, and then, for word $v,w \in A$, define $v<w$ if either $v$ is shorter than $w$, or if $v$ and $w$ have the same length but $v$ precedes $w$ in the lexicographical (i.e. dictionary) ordering of words.

The problem with this is that in general (i.e. for groups with unsolvable word problem) there is no algorithm for computing this normal form.

For a normal form to be useful, it needs to satisfy two properties. Firstly, it should be easy to recognize when a given word is in normal form. Secondly, there should be an algorithm, preferably reasonably efficient, for computing the normal form equivalent of an arbitrary word.

The normal forms that you read about, such as for braid groups and polycyclic groups, satisfy these two properties.