On conjugacy of AltSUB5/SUB-subgroups of Borovik subgroup of group ESUB8/SUB(q)

摘要

Let p ≥ 7 be a prime, q = p n , where n ∈ N, and k be the algebraic closure of the field F q . Let G ～ = E 8 (k) be a simple linear algebraic group of type E 8 over the field k, and σ : G → G be a Steinberg endomorphism of G such that G σ ～ = E 8 (q). Let M ～ = (Alt 5 × Sym 6 ).2 be a Borovik subgroup of the group G and M G σ . An open question is whether the normal Alt 5 -subgroup of M and a diagonal Alt 5 -subgroup of soc(M ) are conjugated in G σ or not. In 1998, D. Frey investigated conjugated classes of Alt 5 -subgroups in E 8 (C). But, description of the classes with zero-dimensional centralizers was not obtained. In particular, it was not clear are Alt 5 -subgroups of a Borovik subgroup of E 8 (C) with zero-dimensional centralizers conjugated in E 8 (C) or not. This problem was solved by G. Lusztig in 2003. Actually, the Lusztig result is more general and concerns regular homorphisms from Alt 5 to connected reductive algebraic group over an algebraically closed field k ′ of characteristic p where p = 0 or p ≥ 7. The Lusztig result implies, in particular, that Alt 5 -subgroups of a Borovik subgroup of E 8 (k ′ ) with zero-dimensional centralizers are conjugated in E 8 (k ′ ). We use the Lusztig result to prove that the normal Alt 5 -subgroup of the group M is conjugated with a diagonal Alt 5 -subgroup of soc(M ) in G σ m where m ≤ 6.