Notes:Free group

Grillet - Abstract Algebra

This is taken from section 6 of chapter 1 starting on page 27.

Reduction

• Let $X$ be a set.
• Let $X'$ be a disjoint set
• Let $A:X\rightarrow X'$ be a bijection, and let $A':\eq A^{-1}:X'\rightarrow X$ be the inverse bijection
• Let $Y:\eq X\cup X'$

Caveat:Apparently we denote $A$ by $x\mapsto x'$ and $A'$ by $y\mapsto y'$ such that $(x')'\eq x$ and $(y')'\eq y$ - I am unsure of this.

Words in the "alphabet" $Y$ are finite, but possibly empty, sequences of elements of $Y$.

Next:

• Let $W$ be the free monoid generated by $Y$, where, as usual, multiplication is concatenation

Reduced word

A word, $a\in W$ with $a\eq(a_1,\ldots,a_n)$ is reduced when:

• $\forall i\in\{1,\ldots,n-1\}[a_{i+1}\neq a_i']$

For example:

1. $(x,y,z)$ - reduced
2. $(x,x,x)$ - reduced
3. $(x,y,y',z)$ - NOT reduced

Reduction deletes subsequences of the form $(a_i,a'_i)$ until a reduced word is reached.

Sequences of reductions

1. We write $a\overset{1}{\rightarrow} b$ if