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 | ||
|---|---|---|---|---|---|
| A5 | Alternating group on 5 letters; = SL2(𝔽4) = L2(5) = L2(4) = icosahedron/dodecahedron rotations; 1st non-abelian simple | 5 | 3+ | A5 | 60,5 |
| d | ρ | Label | ID | ||
|---|---|---|---|---|---|
| GL3(𝔽2) | General linear group on 𝔽23; = Aut(C23) = L3(2) = L2(7); 2nd non-abelian simple | 7 | 3 | GL(3,2) | 168,42 |
| d | ρ | Label | ID | ||
|---|---|---|---|---|---|
| A6 | Alternating group on 6 letters; = PSL2(𝔽9) = L2(9); 3rd non-abelian simple | 6 | 5+ | A6 | 360,118 |