2E6 (mathematics)

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In mathematics, 2E6 is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E6, depending on a quadratic extension of fields KL. Unfortunately the notation for the group is not standardized, as some authors write it as 2E6(K) (thinking of 2E6 as an algebraic group taking values in K) and some as 2E6(L) (thinking of the group as a subgroup of E₆(L) fixed by an outer involution).

Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced independently by Tits (1958) and Steinberg (1959).

Over finite fields[edit]

The group 2E6(q2) has order q36 (q12 − 1) (q9 + 1) (q8 − 1) (q6 − 1) (q5 + 1) (q2 − 1) /(3,q + 1). This is similar to the order q36 (q12 − 1) (q9 − 1) (q8 − 1) (q6 − 1) (q5 − 1) (q2 − 1) /(3,q − 1) of E6(q).

Its Schur multiplier has order (3, q + 1) except for 2E6(22), when it has order 12 and is a product of cyclic groups of orders 2,2,3. One of the exceptional double covers of 2E6(22) is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.

The outer automorphism group has order (3, q + 1) · f where q2pf.

Over the real numbers[edit]

Over the real numbers, 2E6 is the quasisplit form of E6, and is one of the five real forms of E6 classified by E. Cartan. Its maximal compact subgroup is of type F4.

References[edit]