In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem. The complete classification of finite simple groups, completed in 2004, is a major milestone in the history of mathematics. Examples Finite simple groups The cyclic group of congruence classes modulo 3 (see modular arithmetic) is simple. If H is a subgroup of this group, its order (the number of elements) must be a divisor of the order of G which is 3. Since 3 is prime, its only divisors are 1 and 3, so either H is G, or H is the trivial group. On the other hand, the group G = (Z/12Z, +) = Z12 is not simple. The set H of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal. Similarly, the additive group of the integers is not simple; the set of even integers is a non-trivial proper normal subgroup. One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups of prime order. The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is the alternating group A5 of order 60, and every simple group of order 60 is isomorphic to A5. The second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and every simple group of order 168 is isomorphic to PSL(2,7). Infinite simple groups The infinite alternating group, i.e. the group of even finitely supported permutations of the integers, A∞ is simple. This group can be written as the increasing union of the finite simple groups An with respect to standard embeddings . Another family of examples of infinite simple groups is given by PSLn(F), where F is an infinite field and . It is much more difficult to construct finitely generated infinite simple groups. The first existence result is non-explicit; it is due to Graham Higman and consists of simple quotients of the