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There is a slight error in given definition. In given definition, one considers points with value in O, whereas G is only defined over K. One should remark: first that one need to consider a model of G over its ring of integers, and consider G(O); then that the notion of arithmetic group is independant of the choice of a model for G.
In Discrete subgroups of semisimple Lie groups Margulis defines S-arithmetic subgroups for algebraic subgroups of GL(N) over number fields. He defines arithmetic subgroups only for fields over Q (S contaning only the unique archmedean place). He notes that such a (I quote) is intrinsic with respect to the K-structure on H.
Milne's Algebraic Groups and Arithmetic Groups defines an arithmetic subgroup of G(Q) as a subgroup commensurable with the stabilizer of a lattice in a faithfull rational representation. Same definition in Borel Reduction Theory for Arithmetic Groups in PSPUM 9 - Algebraic groups and Discontinuous subgroups.
But in N.D. Alan's The problem of the Maximality of Arithmetic groups (in PSPUM9 too) considers subgroups of G(R) or G(C) commensurable with G(Z) (whatever the integral structure). This is especially relevant for algebraic groups with nontrivial center and when working with arithmetic subgroups of Lie groups (not every connected Lie group, even almost simple, is linear algebraic). —Preceding unsigned comment added by 188.8.131.52 (talk) 19:26, 3 April 2009 (UTC)