of_normal — Mathlib

English

If a is contained in a block B and N ≤ G with N normal, then the orbit of a under N coincides with B when viewed inside the stabilizer context.

Русский

Если a принадлежит блоку B и N нормально, то орбита a под N совпадает с B в контексте стабилизатора.

LaTeX

$$$\\text{orbit}_N(a) = B$ under appropriate identifications$$

Lean4

/-- The orbits of a normal subgroup form a block system -/
@[to_additive /-- The orbits of a normal subgroup form a block system -/
    ]
theorem of_normal {N : Subgroup G} [N.Normal] : IsBlockSystem G (Set.range fun a : X => orbit N a) :=
  by
  constructor
  · apply IsPartition.of_orbits
  · intro b; rintro ⟨a, rfl⟩
    exact .orbit_of_normal a

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