Element of Free Group can be Expressed Uniquely as Finite Product

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