fixingSubgroup_fixedPoints_gc — Mathlib

English

There is a Galois connection between fixingSubgroup M and fixedPoints P α as P ranges over subgroups, and S ranges over subgroups via the dual correspondence.

Русский

Существует связь Галуа между fixingSubgroup M и fixedPoints P α как P простирается по подгруппам, и S через двойственную корреспонденцию.

LaTeX

$$$\\text{GaloisConnection}(OrderDual.toDual \\circ fixingSubgroup M, (\\λ P:Subgroup M. fixedPoints P α) \\circ OrderDual.ofDual)$$$

Lean4

/-- The Galois connection between fixing subgroups and fixed points of a group action -/
@[to_additive]
theorem fixingSubgroup_fixedPoints_gc :
    GaloisConnection (OrderDual.toDual ∘ fixingSubgroup M)
      ((fun P : Subgroup M => fixedPoints P α) ∘ OrderDual.ofDual) :=
  fun _s _P => ⟨fun h s hs p => h p.2 ⟨s, hs⟩, fun h p hp s => h s.2 ⟨p, hp⟩⟩

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