E₆ (mathematics) - Chevalley and Steinberg Groups of Type E₆ and 2E₆

Chevalley and Steinberg Groups of Type E₆ and 2E₆

The groups of type E₆ over arbitrary fields (in particular finite fields) were introduced by Dickson (1901, 1908).

The points over a finite field with q elements of the (split) algebraic group E₆ (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 E₆(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 E₆ (for clarity, this can be written E_6,sc(q) or more rarely and is known as the “universal” Chevalley group of type E₆ over F_q),
• (rarely) the finite group consisting of the points over F_q of the adjoint form of E₆ (for clarity, this can be written E_6,ad(q), and is known as the “adjoint” Chevalley group of type E₆ 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 E₆(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: E₆(q) is simple for any q, E_6,sc(q) is its Schur cover, and E_6,ad(q) lies in its automorphism group; furthermore, when q−1 is not divisible by 3, all three coincide, and otherwise (when q is congruent to 1 mod 3), the Schur multiplier of E₆(q) is 3 and E₆(q) is of index 3 in E_6,ad(q), which explains why E_6,sc(q) and E_6,ad(q) are often written as 3·E₆(q) and E₆(q)·3. From the algebraic group perspective, it is less common for E₆(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_6,sc(q) and E_6,ad(q).

Beyond this “split” (or “untwisted”) form of E₆, there is also one other form of E₆ over the finite field F_q, known as 2E₆, which is obtained by twisting by the non-trivial automorphism of the Dynkin diagram of E₆. Concretely, 2E₆(q), which is known as a Steinberg group, can be seen as the subgroup of E₆(q2) fixed by the composition of the non-trivial diagram automorphism and the non-trivial field automorphism of F_q2. Twisting does not change the fact that the algebraic fundamental group of 2E_6,ad is Z/3Z, but it does change those q for which the covering of 2E_6,ad by 2E_6,sc is non-trivial on the F_q-points. Precisely: 2E_6,sc(q) is a covering of 2E₆(q), and 2E_6,ad(q) lies in its automorphism group; when q+1 is not divisible by 3, all three coincide, and otherwise (when q is congruent to 2 mod 3), the degree of 2E_6,sc(q) over 2E₆(q) is 3 and 2E₆(q) is of index 3 in 2E_6,ad(q), which explains why 2E_6,sc(q) and 2E_6,ad(q) are often written as 3·2E₆(q) and 2E₆(q)·3.

Two notational issues should be raised concerning the groups 2E₆(q). One is that this is sometimes written 2E₆(q2), a notation which has the advantage of transposing more easily to the Suzuki and Ree groups, but the disadvantage of deviating from the notation for the F_q-points of an algebraic group. Another is that whereas 2E_6,sc(q) and 2E_6,ad(q) are the F_q-points of an algebraic group, the group in question also depends on q (e.g., the points over F_q2 of the same group are the untwisted E_6,sc(q2) and E_6,ad(q2)).

The groups E₆(q) and 2E₆(q) are simple for any q, and constitute two of the infinite families in the classification of finite simple groups. Their order is given by the following formula (sequence A008872 in OEIS):

(sequence A008916 in OEIS). The order of E_6,sc(q) or E_6,ad(q) (both are equal) can be obtained by removing the dividing factor gcd(3,q−1) from the first formula (sequence A008871 in OEIS), and the order of 2E_6,sc(q) or 2E_6,ad(q) (both are equal) can be obtained by removing the dividing factor gcd(3,q+1) from the second (sequence A008915 in OEIS).

The Schur multiplier of E₆(q) is always gcd(3,q−1) (i.e., E_6,sc(q) is its Schur cover). The Schur multiplier of 2E₆(q) is gcd(3,q+1) (i.e., 2E_6,sc(q) is its Schur cover) outside of the exceptional case q=2 where it is 22·3 (i.e., there is an additional 22-fold cover). The outer automorphism group of E₆(q) is the product of the diagonal automorphism group Z/gcd(3,q−1)Z (given by the action of E_6,ad(q)), the group Z/2Z of diagram automorphisms, and the group of field automorphisms (i.e., cyclic of order f if q=pf where p is prime). The outer automorphism group of 2E₆(q) is the product of the diagonal automorphism group Z/gcd(3,q+1)Z (given by the action of 2E_6,ad(q)) and the group of field automorphisms (i.e., cyclic of order f if q=pf where p is prime).