E₇ (mathematics) - Chevalley Groups of Type E₇

Chevalley Groups of Type E₇

The points over a finite field with q elements of the (split) algebraic group E₇ (see above), whether of the adjoint (centerless) or simply connected form (its algebraic universal cover), give a finite Chevalley group. This is closely connected to the group written E₇(q), however there is ambiguity in this notation, which can stand for several things:

• the finite group consisting of the points over F_q of the simply connected form of E₇ (for clarity, this can be written E_7,sc(q) and is known as the “universal” Chevalley group of type E₇ over F_q),
• (rarely) the finite group consisting of the points over F_q of the adjoint form of E₇ (for clarity, this can be written E_7,ad(q), and is known as the “adjoint” Chevalley group of type E₇ over F_q), or
• the finite group which is the image of the natural map from the former to the latter: this is what will be denoted by E₇(q) in the following, as is most common in texts dealing with finite groups.

From the finite group perspective, the relation between these three groups, which is quite analogous to that between SL(n, q), PGL(n, q) and PSL(n, q), can be summarized as follows: E₇(q) is simple for any q, E_7,sc(q) is its Schur cover, and the E_7,ad(q) lies in its automorphism group; furthermore, when q is a power of 2, all three coincide, and otherwise (when q is odd), the Schur multiplier of E₇(q) is 2 and E₇(q) is of index 2 in E_7,ad(q), which explains why E_7,sc(q) and E_7,ad(q) are often written as 2·E₇(q) and E₇(q)·2. From the algebraic group perspective, it is less common for E₇(q) to refer to the finite simple group, because the latter is not in a natural way the set of points of an algebraic group over F_q unlike E_7,sc(q) and E_7,ad(q).

As mentioned above, E₇(q) is simple for any q, and it constitutes one of the infinite families addressed by the classification of finite simple groups. Its number of elements is given by the formula (sequence A008870 in OEIS):

The order of E_7,sc(q) or E_7,ad(q) (both are equal) can be obtained by removing the dividing factor gcd(2, q−1) (sequence A008869 in OEIS). The Schur multiplier of E₇(q) is gcd(2, q−1), and its outer automorphism group is the product of the diagonal automorphism group Z/gcd(2, q−1)Z (given by the action of E_7,ad(q)) and the group of field automorphisms (i.e., cyclic of order f if q = pf where p is prime).