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A normal form for free groups and free product of groups represents elements by reduced words. In a free group, adjacent inverse letters cancel; in a free product, alternating nonidentity syllables from different factors are unique. In an amalgamated free product, chosen coset representatives produce the corresponding reduced form relative to the common subgroup.

In $A{\ast }_{C}B$, an alternating product of elements of $A\setminus C$ and $B\setminus C$ is nonidentity. More generally, choosing coset representatives gives a unique reduced normal form.

This is the amalgamated case of normal form for free groups and free product of groups.