By Gundel Klaas, Charles R. Leedham-Green, Wilhelm Plesken

The common subgroup constitution of maximal pro-*p*-subgroups of rational issues of algebraic teams over the *p*-adics and their attribute *p* analogues are investigated. those teams have finite width, i.e. the indices of the sucessive phrases of the reduce crucial sequence are bounded for the reason that they turn into periodic. The richness of the lattice of ordinary subgroups is studied by means of the proposal of obliquity. All simply endless maximal teams with Lie algebras as much as size 14 and so much Chevalley teams and classical teams in attribute zero and *p* are lined. The tools use desktops in small situations and are merely theoretical for the endless sequence utilizing root structures or orders with involutions.

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119. (1) Name of the unitary group, information on the division algebra D, (so that A = Dnx n is described, Q denotes a non-split quaternion algebra). (2) Matrix F (so that together with (1) (A,O) is described), (3) a = number of Wedderburn components of A/radA for the hereditary order A. In case a < n those terms of the sequence (no, . , net-I) which consists of degrees of the components of A/radA over their centres are given, which are not equal to 1. Then A is always of the form described before (VL1 0).

L ~ M1r. Then clearly XCLP centralises an (full) o/1r-flag of M/1rM. Therefore LCLP is contained in an open pro-p-subgroup of U. To prove that LCLP is open, note that CLP : Ugen ----+ A;en is a homeomorphism and therefore maps the open subset L of A-onto the open set LCLP . That LC LP is a group requires L Eland follows now from the next lemma. d. 3) Lemma. Let L E l. Then for any x, y E L there exists a Z E L with XCLPYCLP = ZCLP. , such that Z = L~I Pk(X, y). g. PI = X + Y,P2 = -XY + Y X,P3 = -XY X - Y XY,P4 = (Xy)2 - (Y X)2,P5 = (Xy)2 X + (Y X)2y etc..

P ~ D : PH oP :== (P)o = o-span of the elements of P in A is clearly well defined. It gives rise to a saturation process: For P E P)et P = (l+rad(oP))nU. Clearly, P ~ PEP. A straightforward calculation shows P = P. Define 15 = {PIP E P}. The final map one has to discuss is induced by the first Cayley map CPL. 6) Remark. The map CPL is well defined. Proof. As observed earlier, p =I- 2 implies that PEP lies in Ugen . Obviously PC PL ~ (rad(oP))-, cf. 5)(iii). Let x E (rad(oP)t. Then XCLP = (1 - x)(l + xt l E (1 + rad(oP)) n U, since both 1 + x, 1 - x E 1 + rad(oP) ~ (oP)*.