Groups [and axioms]
Groups have been used implicitly in the context of geometrical symmetries since antiquity. In the late 1700s specific groups began to be studied explicitly, mainly in the context of permutations of roots of polynomials, and notably by Evariste Galois in 1831. General groups were defined by Arthur Cayley around 1850 and their standard axioms became established by the end of the 1800s. The alternate axioms given in the main text are the shortest known. The first for ordinary groups was found by Graham Higman and Bernhard Neumann in 1952; the second by William McCune (using automated theorem proving) in 1992. For commutative (Abelian) groups the first alternate axioms were found by Alfred Tarski in 1938; the second by William McCune (using automated theorem proving) in 1992. In this case it is known that no shorter axioms are possible. (See page 806.) Note that in terms of the \[SmallCircleBar] operator 1 a \[SmallCircleBar] a, OverBar[b] (a \[SmallCircleBar] a) \[SmallCircleBar] b, and a · b a \[SmallCircleBar] ((a \[SmallCircleBar] a) \[SmallCircleBar] b). Ordinary group theory is universal; commutative group theory is not (see page 1159).