= Reduced word in a free group
= Reduced words in a free group
{synonym}
= Reduced word
{synonym}
A <reduced word> contains no adjacent generator and inverse-generator pair. Cancel such pairs to obtain the unique reduced representative in a <free group>. One proof constructs the free <group> itself from <reduced words>, with multiplication given by concatenation followed by cancellation; its action by these operations on words distinguishes distinct reduced representatives.
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