2E6 (mathematics)

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 K⊂L. 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

The group 2E6(q2) has order q36 (q12 − 1) (q9 + 1) (q8 − 1) (q6 − 1) (q5 + 1) (q2 − 1) /(3,q + 1).[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 q=2, i. e. 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 q2 = pf.

Over the real numbers

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

Remarks
  1. Reading example: If q2=22 in 2E6(q2) then q=2 in the order formula q36 (q12 − 1) (q9 + 1) (q8 − 1) (q6 − 1) (q5 + 1) (q2 − 1) /(3,q + 1). However, the group 2E6(22) is sometimes also written 2E6(2) (e. g. in Wilson's Atlas).

References
  • Carter, Roger W. (1989) [1972], Simple groups of Lie type, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-50683-6, MR 0407163
  • Steinberg, Robert (1959), "Variations on a theme of Chevalley", Pacific Journal of Mathematics, 9: 875–891, doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191
  • Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335, archived from the original on 2012-09-10
  • Tits, Jacques (1958), Les "formes réelles" des groupes de type E6, Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e èd. corrigée, Exposé 162, 15, Paris: Secrétariat math'ematique, MR 0106247
  • Robert Wilson: Atlas of Finite Group Representations: Sporadic groups
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