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Involutions of Type G2 Over Fields of Characteristic Two

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Abstract

We continue a study of automorphisms of order 2 of algebraic groups. In particular we look at groups of type G2 over fields k of characteristic two. Let C be an octonion algebra over k; then Aut(C) is a group of type G2 over k. We characterize automorphisms of order 2 and their corresponding fixed point groups for Aut(C) by establishing a connection between the structure of certain four dimensional subalgebras of C and the elements in Aut(C) that induce inner automorphisms of order 2. These automorphisms relate to certain quadratic forms which, in turn, determine the Galois cohomology of the fixed point groups of the involutions. The characteristic two case is unique because of the existence of four dimensional totally singular subalgebras. Over finite fields we show how our results coincide with known results, and we establish a classification of automorphisms of order 2 over infinite fields of characteristic two.

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Authors and Affiliations

  1. Winston-Salem State University, 601 S. Martin Luther King Dr., Winston-Salem, NC, 27110, USA

    John Hutchens

  2. Washington College, 300 Washington Ave., Chestertown, MD, 21620, USA

    Nathaniel Schwartz

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  1. John Hutchens
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  2. Nathaniel Schwartz
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Correspondence to John Hutchens.

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Presented by Jon F. Carlson.

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Hutchens, J., Schwartz, N. Involutions of Type G2 Over Fields of Characteristic Two. Algebr Represent Theor 21, 487–510 (2018). https://doi.org/10.1007/s10468-017-9723-y

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  • DOI: https://doi.org/10.1007/s10468-017-9723-y

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