English
For additive subgroups L1 ≤ L2 of an abelian group E with Q-linear structure, the relative index equals the absolute determinant of a basis matrix change.
Русский
برای گروه ADDITIVE: относительный индекс равен абсолютному детерминанту матрицы перехода между базисами.
LaTeX
$$$$ L_1.\mathrm{relIndex} L_2 = | b_2.det b_1 | $$$$
Lean4
theorem relIndex_eq_abs_det {E : Type*} [AddCommGroup E] [Module ℚ E] (L₁ L₂ : AddSubgroup E) (H : L₁ ≤ L₂) {ι : Type*}
[DecidableEq ι] [Fintype ι] (b₁ b₂ : Basis ι ℚ E) (h₁ : L₁ = .closure (Set.range b₁))
(h₂ : L₂ = .closure (Set.range b₂)) : L₁.relIndex L₂ = |b₂.det b₁| :=
by
rw [AddSubgroup.relIndex_eq_natAbs_det L₁ L₂ H (b₁.addSubgroupOfClosure L₁ h₁) (b₂.addSubgroupOfClosure L₂ h₂),
Nat.cast_natAbs, Int.cast_abs]
change |algebraMap ℤ ℚ _| = _
rw [Basis.det_apply, Basis.det_apply, RingHom.map_det]
congr; ext
simp [Basis.toMatrix_apply]
