Free Group

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

Group presentations

Group Presentation
Free Product with amalgamation