A non-trivial group which has no normal subgroups other than itself and the subgroup consisting of the identity element. A finite abelian group is simple if and only if it is cyclic of prime order. A₅ is the smallest non-abelian simple group. The classification theorem of finite simple groups was a landmark collaborative result of mathematics, occupying much of the 1960s, 1970s, and 1980s. See Jordan-Hölder theorem, sporadic group.