Wondering how to write the different combinations in a free group in a generic way, going off the definition:
The free group $FS$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$.
Wikipedia doesn't offer a generic notation for the set of all expressions/words/terms that can be built from $S$. Instead they just give an example:
For example, if $S = \{a, b, c\}$, then $T = \{a, a^{−1}, b, b^{−1}, c, c^{−1}\}$, and $$ab^{3}c^{-1}ca^{-1}c$$ is a word in $S$.
But in this (arbitrary) paper on free groups found here, they seem to have some sort of generic notation (not sure):
...here we just give a couple of examples of test elements in the free group $F_n = gp\langle x_1,\dotsc,x_n\rangle : x_1^k,x_2^k,\dotsc x_n^k, k \geq 2 [x_1, x_2] \dot [x_3, x_4] \dot \dots \dot [x_{n-1},x_n]...$
What I'm wondering is how to define $ab^{3}c^{-1}ca^{-1}c$ generically, so something where it can handle:
- Arbitrary number of letters (a, b, c, ...)
- The power signs are unique to each letter in the expression.
- $\pm$
Something along the lines of:
$$ \langle x_i^p, \dotsc , x_n^q \mid p,q \in \mathbb{N} \rangle $$
Not sure if that is correct or there is a better more intuitive notation.