In mathematics,
2E
6 is the name of a family of
Steinberg
Steinberg Media Technologies GmbH (trading as Steinberg) is a German musical software and hardware company based in Hamburg. It develops music writing, recording, arranging, and editing software, most notably Cubase, Nuendo, and Dorico. It als ...
or twisted
Chevalley group
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phr ...
s. 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
2E
6(''K'') (thinking of
2E
6 as an
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
...
taking values in ''K'') and some as
2E
6(''L'') (thinking of the group as a subgroup of E
6(''L'') fixed by an outer involution).
Over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
s these groups form one of the 18 infinite families of
finite simple group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s, and were introduced independently by and .
Over finite fields
The group
2E
6(''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 E
6(''q'').
Its Schur multiplier has order (3, ''q'' + 1) except for ''q''=2, i. e.
2''E''
6(2
2), 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(2
2) 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,
2E
6 is the quasisplit form of E
6, and is one of the five real forms of E
6 classified by
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry ...
. Its maximal compact subgroup is of type
F4.
Remarks
References
*
*
*
*
*{{anchor, Wilson
Robert Wilson: ''Atlas of Finite Group Representations: Sporadic groups''
Finite groups
Lie groups