For example, the cyclic group G = Z/3Z 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 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 Z of 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 it is possible to prove that every simple group of order 168 is isomorphic to PSL(2,7).
The finite simple groups are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way prime numbers are the basic building blocks of the integers. This is expressed by the Jordan-Hölder theorem. In a huge collaborative effort, the classification of finite simple groups was accomplished in 1982.
In 1831 Évariste Galois discovered that the alternating groups on five or more points were simple. The next discoveries were by Camille Jordan in 1870. Jordan had found 4 families of simple matrix groups over finite fields of prime order. Later Jordan's results were generalized to arbitrary finite fields by Leonard Dickson. In the process he discovered several new infinite families of groups, now called the classical groups. At about the same time, it was shown that a family of five groups, called the Mathieu groups and first described by Émile Léonard Mathieu in 1861 and 1873, were also simple. Since these five groups were constructed by methods which did not yield infinitely many possibilities, they were called "sporadic" by William Burnside in his 1897 textbook. In 1981 Robert Griess announced that he had constructed Bernd Fischer's "Monster group". The Monster is the largest sporadic simple group having order of 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. Each element of the Monster can be expressed as a 196,883 by 196,883 matrix. Soon after a proof, totaling more than 10,000 pages, was supplied that group theorists had successfully listed all finite simple groups.
Proof: If n is a prime-power, then a group of order n has a nontrivial center and, therefore, is not simple. If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple.