A definable group is a group whose underlying set, group operation, inversion, and identity element are definable by formulas in first-order logic on a fixed structure. The definable sets contained in the group form a Boolean algebra that is preserved by left and right translation.
Definable groups allow algebraic and measure-theoretic properties to be studied by methods from model theory. For example, definable amenability asks whether the group admits a translation-invariant Keisler measure.
