subgroup_eq_top_of_subsingleton — Mathlib

English

The congruence relation defined by the kernel of a homomorphism coincides with the kernel congruence, i.e., the usual kernel relation equals the congruence relation generated by the kernel.

Русский

congruence, определяемое ядром гомоморфизма, совпадает с конгруэнтностью ядра, т.е. обычное ядро = конгруэнтность ядра.

LaTeX

$$QuotientGroup.con f.ker = Con.ker f$$

Lean4

/-- If the quotient by a subgroup gives a singleton then the subgroup is the whole group. -/
@[to_additive /-- If the quotient by an additive subgroup gives a singleton then the additive
    subgroup is the whole additive group. -/
    ]
theorem subgroup_eq_top_of_subsingleton (H : Subgroup G) (h : Subsingleton (G ⧸ H)) : H = ⊤ :=
  top_unique fun x _ => by
    have this : 1⁻¹ * x ∈ H := QuotientGroup.eq.1 (Subsingleton.elim _ _)
    rwa [inv_one, one_mul] at this

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