In a free group, a reduced word has no adjacent inverse letters. Successively cancelling such pairs gives the unique reduced representative in the chosen free generators. In a free product, a reduced word instead uses nonidentity factor elements as syllables in a free product, with consecutive syllables in a free product from different factors; the alternating syllables in a free product are unique.
For an amalgamated free product , the usual reduced form relative to a common subgroup requires the maps and to be injective. Chosen representatives of cosets then distinguish the alternating factors and the remaining common-subgroup element. The exact form depends on the chosen left/right representative convention.
For general group pushouts defined by arbitrary homomorphisms, factors can collapse and this embedded-subgroup normal form is unavailable. For example, take , map identically into , and map it trivially into . The pushout kills all of and is isomorphic to , so words in nonidentity elements of cannot retain the usual nontrivial-factor normal form.
Every element of a free product has a unique expression in which each is a nonidentity element of one factor and adjacent syllables belong to different factors. The identity corresponds to the empty word.
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In group theory, the concept of "normal form" can refer to a variety of representations that provide a canonical way to express elements in certain types of groups, particularly free groups and free products of groups. ### Normal Form for Free Groups A **free group** is a group where the elements can be represented as reduced words over a set of generators, with no relations other than those that are necessary to satisfy the group axioms (e.g., inverses for each generator).