Difference between revisions of "Notes:Free group"

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Grillet - Abstract Algebra

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

Reduction

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

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

Words in the "alphabet" [ilmath]Y[/ilmath] are finite, but possibly empty, sequences of elements of [ilmath]Y[/ilmath].

Next:

• Let [ilmath]W[/ilmath] be the free monoid generated by [ilmath]Y[/ilmath], where, as usual, multiplication is concatenation^{[Note 1]}

Reduced word

A word, [ilmath]a\in W[/ilmath] with [ilmath]a\eq(a_1,\ldots,a_n)[/ilmath] is reduced when:

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

For example:

1. [ilmath](x,y,z)[/ilmath] - reduced
2. [ilmath](x,x,x)[/ilmath] - reduced
3. [ilmath](x,y,y',z)[/ilmath] - NOT reduced

Reduction deletes subsequences of the form [ilmath](a_i,a'_i)[/ilmath] until a reduced word is reached.

Sequences of reductions

1. We write [ilmath]a\overset{1}{\rightarrow} b[/ilmath] if

Notes

1. ↑ Obviously, concatenation of finite sequences [ilmath]a:\eq(a_1,\ldots,a_\ell)[/ilmath] and [ilmath]b:\eq(b_1,\ldots,b_m)[/ilmath] is:
   • [ilmath]a\cdot b:\eq(a_1,\ldots,a_\ell,b_1,\ldots,b_m)[/ilmath]