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 isomorphism^{} 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

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The first trivial example of simple groups^{} are the cyclic groups^{} of prime (http://planetmath.org/Prime) order. It is not difficult to see (say, by Cauchy’s theorem) that these are the only abelian^{} simple groups.
Alternating groups

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The alternating group^{} on $n$ symbols is the set of all even permutations^{} of ${S}_{n}$, the symmetric group^{} on $n$ symbols. It is usually denoted by ${A}_{n}$, or sometimes by $\mathrm{Alt}(n)$. This is a normal subgroup^{} of ${S}_{n}$, namely the kernel of the homomorphism^{} that sends every even permutation to $1$ and the odd permutations to $1$. Because every permutation^{} is either even or odd, and there is a bijection between the two (multiply every even permutation by a transposition^{}), the index of ${A}_{n}$ in ${S}_{n}$ is $2$. ${A}_{3}$ is simple because it only has three elements, and the simplicity of ${A}_{n}$ for $n\ge 5$ 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 commutators^{}, are therefore uninteresting in ${A}_{n}$ for $n\ge 3$. Specifically, $[{A}_{n},{A}_{n}]={A}_{n}$ and ${A}_{n}$ has trivial center for $n\ge 3$.
Groups of Lie type
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Other groups of Lie type.
Sporadic groups
There are twentysix sporadic groups (no more, no less!) that do not fit into any of the infinite^{} sequences^{} of simple groups considered above. These often arise as the group of automorphisms of strongly regular graphs.

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Mathieu groups^{}.

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Janko groups^{}.

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The baby monster.

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The monster.
Title  examples of finite simple groups 

Canonical name  ExamplesOfFiniteSimpleGroups 
Date of creation  20130322 13:07:54 
Last modified on  20130322 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 