Editing Building (mathematics) (section)

===Automorphisms===
Tits proved that any label-preserving [[automorphism]] of the affine building arises from an element of {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}. Since automorphisms of the building permute the labels, there is a natural homomorphism

:{{math|Aut ''X'' → ''S''<sub>''n''</sub>}}.

The action of {{math|GL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} gives rise to an [[cyclic permutation|{{mvar|n}}-cycle]]&nbsp;{{mvar|τ}}. Other automorphisms of the building arise from [[outer automorphism]]s of {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}} associated with automorphisms of the [[Dynkin diagram]]. Taking the standard symmetric bilinear form with orthonormal basis {{math|''v''<sub>''i''</sub>}}, the map sending a lattice to its dual lattice gives an automorphism whose square is the identity, giving the permutation {{mvar|σ}} that sends each label to its negative modulo {{mvar|n}}. The image of the above homomorphism is generated by {{mvar|σ}} and {{mvar|τ}} and is isomorphic to the [[dihedral group]] {{math|''D''<sub>''n''</sub>}} of order {{math|2''n''}}; when {{math|''n'' {{=}} 3}}, it gives the whole of {{math|''S''<sub>3</sub>}}.

If {{mvar|E}} is a finite [[Galois extension]] of {{math|'''Q'''<sub>''p''</sub>}} and the building is constructed from {{math|SL<sub>''n''</sub>(''E'')}} instead of {{math|SL<sub>''n''</sub>('''Q'''<sub>''p''</sub>)}}, the [[Galois group]] {{math|Gal(''E'' / '''Q'''<sub>''p''</sub>)}} will also act by automorphisms on the building.