کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4584764 | 1630503 | 2014 | 25 صفحه PDF | دانلود رایگان |
Let k be a field of characteristic different from 2 and 3. Let G be a simple group of type F4F4 or G2G2 defined over k . In this paper we discuss embeddings of connected simple algebraic groups of type A1A1 and A2A2 in G in terms of the mod-2 Galois cohomological invariants attached to these groups. We prove that k -groups of type F4F4 (resp. G2G2) arising from division algebras are generated by k -subgroups of type A2A2 (resp. A1A1) (see Theorem 3.11 and Theorem 4.1). We derive a necessary and sufficient condition for an Albert algebra to have zero f5f5 invariant (Theorem 3.4). Further, for a k-group G of type F4F4, we derive a condition necessary for a k-group H of type A1A1 or A2A2 to embed in G over k. We prove that in order to embed H in G over k, the mod-2 invariant of H must divide f5(G)f5(G) (see Section 2 and Remark 2.8 and Remark 2.3 for definition of invariants). Along similar lines, we derive a condition for a k-group H of type A2A2 to embed in a k-group G of type G2G2. We prove that H embeds in G over k , if and only if f3(H)=f3(G)f3(H)=f3(G) (Theorem 4.4). Next we derive a necessary and sufficient condition for a k-group H of type A1A1 to embed in a k-group G of type G2G2. If G=Aut(C)G=Aut(C) is a group of type G2G2 over k for an octonion algebra C over k, then this condition provides a natural bijection between k -conjugacy classes of involutions in G(k)G(k) and isometry classes of 2-fold Pfister divisors of the Pfister form nCnC, the norm form of the octonion algebra C (Section 4, Proposition 4.2).
Journal: Journal of Algebra - Volume 411, 1 August 2014, Pages 312–336