2E6 (mathematics)

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 Ã¢ÂˆÂ’ 1) (q⁹ + 1) (q⁸ Ã¢ÂˆÂ’ 1) (q⁶ Ã¢ÂˆÂ’ 1) (q⁵ + 1) (q² Ã¢ÂˆÂ’ 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 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) Ã‚· 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

• Robert Wilson: Atlas of Finite Group Representations: Sporadic groups