Simple group
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In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself.
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.
The famous theorem of Feit and Thompson states that every group of odd order is solvable. Therefore every finite simple group has even order unless it is cyclic of prime order.
Simple groups of infinite order also exist: simple Lie groups and the infinite Thompson groups T and V are examples of these.
The Schreier conjecture asserts that the group of outer automorphisms of every finite simple group is solvable. This can be proved using the classification theorem.
