commensurable_conj — Mathlib

English

If H and K are commensurable, then their conjugates under any conjugation by g are also commensurable.

Русский

Если H и K комменсурируемы, то их конъюгированные по любому g конгруации also комменсурируемы.

LaTeX

$$∀ g, H.Commensurable K → (instHSMul.hSMul g H).Commensurable (instHSMul.hSMul g K)$$

Lean4

theorem commensurable_conj {H K : Subgroup G} (g : ConjAct G) : Commensurable H K ↔ Commensurable (g • H) (g • K) :=
  and_congr (not_iff_not.mpr (Eq.congr_left (Cardinal.toNat_congr (quotConjEquiv H K g))))
    (not_iff_not.mpr (Eq.congr_left (Cardinal.toNat_congr (quotConjEquiv K H g))))

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