Simple groups

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, a normal subgroup and the quotient group, and the process can be repeated. If the group is finite, then eventually one arrives at uniquely determined simple groups by the Jordan–Hölder theorem. The complete classification of finite simple groups, completed in 2008, is a major milestone in the history of mathematics.

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).

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