# Automatic group

In mathematics, an **automatic group** is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph of the group. That is, they can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator.^{[1]}

More precisely, let *G* be a group and *A* be a finite set of generators. Then an *automatic structure* of *G* with respect to *A* is a set of finite-state automata:^{[2]}

The property of being automatic does not depend on the set of generators.^{[3]}

A group is **biautomatic** if it has two multiplier automata, for left and right multiplication by elements of the generating set, respectively. A biautomatic group is clearly automatic.^{[7]}

The idea of describing algebraic structures with finite-automata can be generalized from groups to other structures.^{[9]} For instance, it generalizes naturally to automatic semigroups.^{[10]}