Max-Albert Knus, Jean-Pierre Tignol: Triality and Algebraic Groups of Type 3D4,

Submission: 2014, Sep 5

We determine which simple algebraic groups of type 3D4 over arbitrary fields of characteristic different from 2 admit outer automorphisms of order 3, and classify these automorphisms up to conjugation. The criterion is formulated in terms of a representation of the group by automorphisms of a trialitarian algebra: outer automorphisms of order 3 exist if and only if the algebra is the endomorphism algebra of an induced cyclic composition; their conjugacy classes are in one-to-one correspondence with isomorphism classes of symmetric compositions from which the induced cyclic composition stems.

2010 Mathematics Subject Classification: 20G15, 11E72, 17A75

Keywords and Phrases: Algebraic group of outer type 3D4, triality, outer automorphism of order 3, composition algebra, symmetric composition, cyclic composition, octonions, Okubo algebra

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