G is a finite group,. Define R = R(G) = intersection{K normal in G such that G/K
ID: 3081907 • Letter: G
Question
G is a finite group,. Define R = R(G) = intersection{K normal in G such that G/K is solvable}. Define S = S(G) = pi{H normal in G such that H is solvable}. If a map alpha from G to G_1 is a group homomorphism, show that alpha[R(G)] is contained in R(G_1) and alpha[S(G)] is contained in S(G_1).Explanation / Answer
Consider a smooth connected solvable group G over a field k. If k is algebraically closed then G = T ltimes mathscr{R}u(G) for any maximal torus T of G [borelag] . Over more general k, an analogous such structure can fail to exist. For example, consider an imperfect field k of characteristic p > 0 and an element a ? k - kp, so k’ := k(a1/p) is a degree-p purely inseparable extension of k. Note that k’s := k’ ?k ks = ks(a1/p) is a separable closure of k’, and k’sp ? ks. The affine Weil restriction G = Rk’/k(Gm) is an open subscheme of Rk’/k(A1k’) = Apk, so it is a smooth connected affine k-group of dimension p > 1. Loosely speaking, G is “k’× viewed as a k-group”. More precisely, for k-algebras R we have G(R) = (k’ ?k R)× functorially in R. (See Exercise U.4 for a treatment of Weil restriction in the affine case.) The commutative k-group G contains an evident 1-dimensional torus T ? Gm corresponding to the subgroup R× ? (k’ ?k R)×, and G/T is unipotent because (G/T)(ks) = (k’s)×/(ks)× is p-torsion. In particular, T is the unique maximal torus of G. Since the group G(ks) = k’s× has no nontrivial p-torsion, G contains no nontrivial unipotent smooth connected k-subgroup. Thus, G is a commutative counterexample over k to the analogue of the semi-direct product structure for connected solvable smooth affine groups over k. The appearance of imperfect fields in the preceding counterexample is essential. To explain this, recall Grothendieck’s theorem that over a general field k, if S is a maximal k-torus in a smooth affine k-group H then Sk is maximal in Hk. (This theorem is an application of [sga3] to the smooth affine k-group ZH(S), since a “maximal torus” over k in the sense of [sga3] is defined to be a k-torus that is maximal after scalar extension to k. For another proof, see [sga3notes] .) Thus, by the conjugacy of maximal tori in Gk, G = T ltimes U for a k-torus T and a unipotent smooth connected normal k-subgroup U ? G if and only if the subgroup mathscr{R}u(Gk) ? Gk is defined over k (i.e., descends to a k-subgroup of G). In such cases, the semi-direct product structure holds for G over k using any maximal k-torus T of G (and U is unique: it must be a k-descent of mathscr{R}u(Gk)). If k is perfect then by Galois descent we may always descend mathscr{R}u(Gk) to a k-subgroup of G. The main challenge is the case of imperfect k. In these notes, we explain Tits’ structure theory for unipotent smooth connected groups over general fields of positive characteristic (especially imperfect fields), which builds on earlier work of Rosenlicht [rosenlicht] and concerns the structure of smooth connected unipotent groups as well as torus actions on such groups over an arbitrary ground field of positive characteristic. These results on unipotent groups were presented by Tits in a course at Yale University in 1967, and lecture notes [titsyale] for that course were circulated but never published. Much of the course was concerned with general results on linear algebraic groups that are available now in many standard references (such as [borelag] , [humphreys] , and [springer] ). The original account (with proofs) of Tits’ structure theory of unipotent groups is his unpublished Yale lecture notes, and a summary of the results is given in [oesterle] . We use Tits’ work to establish a general structure theorem for solvable smooth connected groups that replaces (and generalizes) the semi-direct product structure over perfect k. The bulk of the work is in the unipotent case, for which our exposition is an improvement of [pred] via simplifications in some proofs. (This simplified treatment of Tits’ work will also appear in Appendix B of the second edition of [pred] .) In some parts we have simply reproduced arguments from Tits’ lecture notes. Throughout the discussion below, k is an arbitrary field with characteristic p > 0.
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