k1

K1 is a designation that appears across different disciplines, and there is no single, universal meaning. It is most prominently encountered in mathematics, where it denotes a fundamental invariant in algebraic K-theory, and in the branding of a major kickboxing organization. The term also appears in a range of other contexts, including transportation, road designations, and product naming.

In mathematics, K1 refers to the first K-group in algebraic K-theory. For a ring R, K1(R) is defined as GL(R)ab, the abelianization of the infinite general linear group GL(R). For a field F, K1(F) is isomorphic to F×, the multiplicative group of nonzero elements. K1 is connected to determinants, automorphisms, and structural properties of rings, and it sits at the base of Quillen's higher K-theory.

In the realm of combat sports, K-1 denotes a prominent kickboxing organization established in 1993 by Kazuyoshi Ishii. It organized the K-1 World Grand Prix events and played a key role in popularizing modern kickboxing through its rule system and international events.

Other uses include occasional road or route designations in various countries and model codes in consumer products, where the K1 label serves as a simple alphanumeric identifier.