Simple group

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A simple group is a nontrivial group that has no nontrivial normal subgroups. If a group isn’t simple, you can find a nontrivial normal subgroup and look at the quotient group; this process can continue until only simple pieces are left. For finite groups, the Jordan–Hölder theorem says these simple pieces are determined up to order.

Examples and what they mean
- Simple abelian groups: The only simple abelian groups are cyclic groups of prime order. For example, Z3 is simple because its only subgroups have order 1 or 3. Z12 is not simple because it has a normal subgroup of order 3.
- Infinite abelian groups: The additive group of integers Z is not simple because the even integers form a nontrivial normal subgroup.

Nonabelian simple groups
- The smallest nonabelian simple group is A5, which has 60 elements. Any simple group of order 60 is isomorphic to A5.
- The next famous one is PSL(2,7) with 168 elements; any simple group of order 168 is isomorphic to PSL(2,7).
- There are infinite families of simple groups too, such as A∞ (the group of even, finitely supported permutations of the integers) and PSLn(F) for infinite fields F with n ≥ 2.
- There are also interesting finitely presented infinite simple groups, such as the Thompson groups.

Finite vs infinite and classification
- There are many finite simple groups, and they were shown to fall into a big, finite list: 18 families plus 26 exceptional “sporadic” groups. The Monster group is the largest sporadic simple group.
- A key result (Feit–Thompson) says every group of odd order is solvable, so any finite simple group that isn’t cyclic of prime order has even order.

Why simple groups matter
- Simple groups act like the basic building blocks for all finite groups, in a way similar to how prime numbers are the building blocks for integers. The Jordan–Hölder theorem formalizes this idea by showing the simple factors in any finite group are essentially unique.