map_normalizer_eq_of_bijective — Mathlib

English

Kernel of the product map equals the product of kernels (repeat statement).

Русский

Ядро отображения на произведение равно произведению ядер (повтор).

LaTeX

$$$ \\ker (\\\\mathrm{prodMap}(f,g)) = \\ker f \\\\mathrm{.prod} \\\\ker g $$$

Lean4

/-- The image of the normalizer is equal to the normalizer of the image of a bijective
  function. -/
@[to_additive /-- The image of the normalizer is equal to the normalizer of the image of a bijective
            function. -/
    ]
theorem map_normalizer_eq_of_bijective (H : Subgroup G) {f : G →* N} (hf : Function.Bijective f) :
    H.normalizer.map f = (H.map f).normalizer :=
  map_equiv_normalizer_eq H (MulEquiv.ofBijective f hf)

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