From Surf Wiki (app.surf) — the open knowledge base
2E6 (mathematics)
Family of groups in group theory
Family of groups in group theory
In mathematics, 2E6 is 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 and .
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).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). 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 = p**f.
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
References
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
Ask Mako anything about 2E6 (mathematics) — get instant answers, deeper analysis, and related topics.
Research with MakoFree with your Surf account
Create a free account to save articles, ask Mako questions, and organize your research.
Sign up freeThis content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.
Report