Abstract
LetG be an algebraic group over a fieldk. We callg eG(k) real ifg is conjugate tog −1 inG(k). In this paper we study reality for groups of typeG 2 over fields of characteristic different from 2. LetG be such a group overk. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element inG(k) is real if and only if it is a product of two involutions inG(k). Every unipotent element inG(k) is a product of two involutions inG(k). We discuss reality forG 2 over special fields and construct examples to show that reality fails for semisimple elements inG 2 over ℚ and ℚp. We show that semisimple elements are real forG 2 overk withcd(k) ≤ 1. We conclude with examples of nonreal elements inG 2 overk finite, with characteristick not 2 or 3, which are not semisimple or unipotent.
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