Normal form for free groups and free product of groups

In mathematics, particularly in combinatorial group theory, a normal form for a free group over a set of generators or for a free product of groups is a representation of an element by a simpler element, the element being either in the free group or free products of group. In case of free group these simpler elements are reduced words and in the case of free product of groups these are reduced sequences. The precise definitions of these are given below. As it turns out, for a free group and for the free product of groups, there exists a unique normal form i.e each element is representable by a simpler element and this representation is unique. This is the Normal Form Theorem for the free groups and for the free product of groups. The proof here of the Normal Form Theorem follows the idea of Artin and van der Waerden.

Normal Form for Free Groups

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Let   be a free group with generating set  . Each element in   is represented by a word   where  

Definition. A word is called reduced if it contains no string of the form  

Definition. A normal form for a free group   with generating set   is a choice of a reduced word in   for each element of  .

Normal Form Theorem for Free Groups. A free group has a unique normal form i.e. each element in   is represented by a unique reduced word.

Proof. An elementary transformation of a word   consists of inserting or deleting a part of the form   with  . Two words   and   are equivalent,  , if there is a chain of elementary transformations leading from   to  . This is obviously an equivalence relation on  . Let   be the set of reduced words. We shall show that each equivalence class of words contains exactly one reduced word. It is clear that each equivalence class contains a reduced word, since successive deletion of parts   from any word   must lead to a reduced word. It will suffice then to show that distinct reduced words   and   are not equivalent. For each   define a permutation   of   by setting   if   is reduced and   if  . Let   be the group of permutations of   generated by the  . Let   be the multiplicative extension of   to a map  . If   then  ; moreover   is reduced with   It follows that if   with   reduced, then  .

Normal Form for Free Products

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Let   be the free product of groups   and  . Every element   is represented by   where   for  .

Definition. A reduced sequence is a sequence   such that for   we have   and   are not in the same factor   or  . The identity element is represented by the empty set.

Definition. A normal form for a free product of groups is a representation or choice of a reduced sequence for each element in the free product.

Normal Form Theorem for Free Product of Groups. Consider the free product   of two groups   and  . Then the following two equivalent statements hold.
(1) If  , where   is a reduced sequence, then   in  
(2) Each element of   can be written uniquely as   where   is a reduced sequence.

Proof

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Equivalence

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The fact that the second statement implies the first is easy. Now suppose the first statement holds and let:

 

This implies

 

Hence by first statement left hand side cannot be reduced. This can happen only if   i.e.   Proceeding inductively we have   and   for all   This shows both statements are equivalent.

Proof of (2)

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Let W be the set of all reduced sequences in AB and S(W) be its group of permutations. Define φ : AS(W) as follows:

 

Similarly we define ψ : BS(W).

It is easy to check that φ and ψ are homomorphisms. Therefore by universal property of free product we will get a unique map φψ : ABS(W) such that φψ (id)(1) = id(1) = 1.

Now suppose   where   is a reduced sequence, then   Therefore w = 1 in AB which contradicts n > 0.

References

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  • Lyndon, Roger C.; Schupp, Paul E. (1977). Combinatorial Group Theory. Springer. ISBN 978-3-540-41158-1..