Element of Free Group can be Expressed Uniquely as Finite Product
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Theorem
Let $G$ be a free group.
Let $x \in G$ be an element of $G$ other than the identity element.
Then $x$ can be expressed uniquely in the form:
- $a^\alpha b^\beta \dotsm r^\sigma$
where:
- adjacent elements $a, b, \ldots, r$ of $a^\alpha b^\beta \dotsm r^\sigma$ are distinct elements of $G$
- $\alpha, \beta, \ldots, \sigma$ are non-zero integers.
Proof
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Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): free group
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): free group