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 and .

Over finite fields

The group ²E₆(''q''²) has order
''q''³⁶
(''q''¹² − 1)
(''q''⁹ + 1)
(''q''⁸ − 1)
(''q''⁶ − 1)
(''q''⁵ + 1)
(''q''² − 1)
/(3,''q'' + 1).Reading example: If ''q''²=2² in ²E₆(''q''²) then ''q''=2 in the order formula ''q''³⁶
(''q''¹² − 1)
(''q''⁹ + 1)
(''q''⁸ − 1)
(''q''⁶ − 1)
(''q''⁵ + 1)
(''q''² − 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 ''q''³⁶
(''q''¹² − 1)
(''q''⁹ − 1)
(''q''⁸ − 1)
(''q''⁶ − 1)
(''q''⁵ − 1)
(''q''² − 1)
/(3,''q'' − 1)
of E₆(''q'').

Its Schur multiplier has order (3, ''q'' + 1) except for ''q''=2, i. e. ²''E''₆(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 ²''E''₆(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 ''q''² = ''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

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*{{anchor, Wilson
Robert Wilson: ''Atlas of Finite Group Representations: Sporadic groups''

Finite groups
Lie groups