examples of finite simple groups

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All groups considered here are finite.

It is now widely believed that the classification of all finite simple groups up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is finished. The proof runs for at least 10,000 printed pages, and as of the writing of this entry, has not yet been published in its entirety.

Abelian groups

Alternating groups

  • The alternating groupMathworldPlanetmath on n symbols is the set of all even permutationsMathworldPlanetmath of Sn, the symmetric groupMathworldPlanetmathPlanetmath on n symbols. It is usually denoted by An, or sometimes by Alt(n). This is a normal subgroupMathworldPlanetmath of Sn, namely the kernel of the homomorphismMathworldPlanetmathPlanetmathPlanetmath that sends every even permutation to 1 and the odd permutations to -1. Because every permutationMathworldPlanetmath is either even or odd, and there is a bijection between the two (multiply every even permutation by a transpositionMathworldPlanetmath), the index of An in Sn is 2. A3 is simple because it only has three elements, and the simplicity of An for n5 can be proved by an elementary argument. The simplicity of the alternating groups is an important fact that Évariste Galois required in order to prove the insolubility by radicals of the general polynomial of degree higher than four. It is worth noting that some common sources of normal subgroups, namely centers and commutatorsPlanetmathPlanetmath, are therefore uninteresting in An for n3. Specifically, [An,An]=An and An has trivial center for n3.

Groups of Lie type

Sporadic groups

There are twenty-six sporadic groups (no more, no less!) that do not fit into any of the infiniteMathworldPlanetmath sequencesPlanetmathPlanetmath of simple groups considered above. These often arise as the group of automorphisms of strongly regular graphs.

Title examples of finite simple groups
Canonical name ExamplesOfFiniteSimpleGroups
Date of creation 2013-03-22 13:07:54
Last modified on 2013-03-22 13:07:54
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 15
Author mathcam (2727)
Entry type Example
Classification msc 20A05
Classification msc 20E32
Related topic ExamplesOfGroups
Related topic SimplicityOfA_n
Related topic JankoGroups
Defines alternating group