Group of Lie Type - Notation Issues

Notation Issues

Unfortunately there is no standard notation for the finite groups of Lie type, and the literature contains dozens of incompatible and confusing systems of notation for them.

• The simple group PSL(n, q) is not usually the same as the group PSL(n, F_q) of F_q-valued points of the algebraic group PSL(n). The problem is that a surjective map of algebraic groups such as SL(n) → PSL(n) does not necessarily induce a surjective map of the corresponding groups with values in some (non algebraically closed) field. There are similar problems with the points of other algebraic groups with values in finite fields.

• The groups of type A_n−1 are sometimes denoted by PSL(n, q) (the projective special linear group) or by L(n, q).

• The groups of type C_n are sometimes denoted by Sp(2n, q) (the symplectic group) or (confusingly) by Sp(n, q).

• The notation for groups of type D_n ("orthogonal" groups) is particularly confusing. Some symbols used are O(n, q), O−(n, q), PSO(n, q), Ω_n(q), but there are so many conventions that it is not possible to say exactly what groups these correspond to without it being specified explicitly. The source of the problem is that the simple group is not the orthogonal group O, nor the projective special orthogonal group PSO, but rather a subgroup of PSO, which accordingly does not have a classical notation. A particularly nasty trap is that some authors, such as the ATLAS, use O(n, q) for a group that is not the orthogonal group, but the corresponding simple group. The notation Ω, PΩ was introduced by Jean Dieudonné, though his definition is not simple for n ≤ 4 and thus the same notation may be used for a slightly different group, which agrees in n ≥ 5 but not in lower dimension.

• For the Steinberg groups, some authors write 2A_n(q2) (and so on) for the group that other authors denote by 2A_n(q). The problem is that there are two fields involved, one of order q2, and its fixed field of order q, and people have different ideas on which should be included in the notation. The "2A_n(q2)" convention is more logical and consistent, but the "2A_n(q)" convention is far more common and is closer to the convention for algebraic groups.

• Authors differ on whether groups such as A_n(q) are the groups of points with values in the simple or the simply connected algebraic group. For example, A_n(q) may mean either the special linear group SL(n+1, q) or the projective special linear group PSL(n+1, q). So 2A₂(4) may be any one of 4 different groups, depending on the author.