A finite group G is *simple* if it has only two normal subgroups - the trivial group
and G itself. Simple groups are building blocks for all finite groups,
via extensions, and they are divided into cyclic groups of prime order and the
*non-abelian simple groups*, the 'interesting ones', listed below.
For small orders they are all alternating or linear.

See also almost simple, quasisimple and non-soluble groups.

d | ρ | Label | ID | ||
---|---|---|---|---|---|

A_{5} | Alternating group on 5 letters; = SL_{2}(𝔽_{4}) = L_{2}(5) = L_{2}(4) = icosahedron/dodecahedron rotations; 1^{st} non-abelian simple | 5 | 3+ | A5 | 60,5 |

d | ρ | Label | ID | ||
---|---|---|---|---|---|

GL_{3}(𝔽_{2}) | General linear group on 𝔽_{2}^{3}; = Aut(C_{2}^{3}) = L_{3}(2) = L_{2}(7); 2^{nd} non-abelian simple | 7 | 3 | GL(3,2) | 168,42 |

d | ρ | Label | ID | ||
---|---|---|---|---|---|

A_{6} | Alternating group on 6 letters; = PSL_{2}(𝔽_{9}) = L_{2}(9); 3^{rd} non-abelian simple | 6 | 5+ | A6 | 360,118 |

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