For nonempty $S\subset G$, we can form infinitely many words in $\langle S\rangle$, but often there are many words that represent the same group element. We can partition the collection of words in the alphabet $S$ into equivalence classes based on which group element a word represent.
Need to make a comment about how we are being slightly abusive in our notation and how we identify a word with a group element and an equivalence class with a group element.
dcernst
commented
4 years ago
For nonempty $S\subset G$, we can form infinitely many words in $\langle S\rangle$, but often there are many words that represent the same group element. We can partition the collection of words in the alphabet $S$ into equivalence classes based on which group element a word represent.
Need to make a comment about how we are being slightly abusive in our notation and how we identify a word with a group element and an equivalence class with a group element.