Group Theory/Free products and amalgamated sums

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Definition (reduced word):

Let be any set, and define the set to be the set of formal inverses to the elements of ; that is, , so that ; for example, we could define . Let denote the empty tuple. Then a reduced word over is either

  1. the empty tuple , or
  2. a finite tuple of elements of such that whenever are two adjacent elements, then neither and nor and .

Definition (empty word):

The empty tuple is also called the empty word

Proposition (reduction of tuples to reduced words):

Let be any set, and let be the set of formal inverses. Suppose that is any tuple (not necessarily a reduced word). Then in finitely many steps, one may obtain a reduced word from by removing adjacent elements such that either and or and .

Proof: This follows immediately since the length of the tuple is an integer, which is reduced by 2 whenever adjacent elements that contradict the definition of a reduced word are eliminated. Doing this elimination repeatedly until it is no longer possible will hence lead to a reduced word in a finite number of steps.

Note that when is odd, then the reduced word obtained in this way will not be the empty tuple. Otherwise, the empty tuple may result.

Definition (free group):

Let be any set. Then the free group over is defined to be the group whose elements are the reduced words over and whose group operation is given by first concatenation and then reduction to a reduced word.

Proposition (the free group is a group):

Let be a set. Then is a group.

Proof: The empty tuple serves as an identity. Associativity holds because if are three reduced words, then

Finally, whenever is a reduced word, we claim by induction on that it has an inverse. Certainly the empty word has Indeed, suppose that ; then , which has an inverse by the induction hypothesis, so that by associativity is an inverse of .


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  1. Prove that when is a set such that , then is not an abelian group.