G₂ (mathematics) - Finite Groups

Finite Groups

The group G₂(q) is the points of the algebraic group G₂ over the finite field F_q. These finite groups were first introduced by Leonard Eugene Dickson in Dickson (1901) for odd q and Dickson (1905) for even q. The order of G₂(q) is q6(q6−1)(q2−1). When q≠2, the group is simple, and when q = 2, it has a simple subgroup of index 2 isomorphic to 2A₂(32). The J₁ was first constructed as a subgroup of G₂(11). Ree (1960) introduced twisted Ree groups 2G₂(q) of order q3(q3+1)(q−1) for q=32n+1 an odd power of 3.