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3D4
In mathematics, the Steinberg triality groups of type 3D4 form a family of Steinberg or twisted Chevalley groups. They are quasi-split forms of D4, depending on a cubic Galois extension of fields ''K'' ⊂ ''L'', and using the triality automorphism of the Dynkin diagram D4. Unfortunately the notation for the group is not standardized, as some authors write it as 3D4(''K'') (thinking of 3D4 as an algebraic group taking values in ''K'') and some as 3D4(''L'') (thinking of the group as a subgroup of D4(''L'') fixed by an outer automorphism of order 3). The group 3D4 is very similar to an orthogonal or spin group in dimension 8.
Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced by .
==Construction==
The simply connected split algebraic group of type D4 has a triality automorphism σ of order 3 coming from an order 3 automorphism of its Dynkin diagram. If ''L'' is a field with an automorphism τ of order 3, then this induced an order 3 automorphism τ of the group D4(''L''). The group 3D4(''L'') is the subgroup of D4(''L'') of points fixed by στ. It has three 8-dimensional representations over the field ''L'', permuted by the outer automorphism τ of order 3.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
■ウィキペディアで「3D4」の詳細全文を読む
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