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Normal form for free groups and free product of groups

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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..