set “alphabet”
also set of symbols (this is just a label, its not an actual inverse yet)
Word in the alphabet finite sequence
A word is reduced if it has no subwords of form or
Elementary reduction of a word
Replace with
Similarly for
Define the free group on the alphabet , the set of reduced words in (including ). Group operation: concatenate words
then apply elementary reductions iteratively.
well defined and associative in handout ! (also ORW notes)
Let with
Universal property of free groups
For any group , the function:
is a bijection
Proof
Given set
If is not reduced: say it contains subword then the image contains
So if two words related by element reduction, then has the same image
Group operation on is concatenat+reduct, so is homomorphism