Theorems · Theorem · group theory
Subgroup.fg_iff_submonoid_fg
∀ {G : Type u_3} [inst : Group G] (P : Subgroup G), P.FG ↔ P.FGA subgroup is finitely generated if and only if it is finitely generated as a submonoid.
- Defined in
- Mathlib.GroupTheory.Finiteness
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Group
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Finsetproof · cited by 13,712
- SetLike.coeproof · cited by 8,199
- Groupstatement and proof · cited by 6,238
- Subgroupstatement and proof · cited by 3,593
- Submonoidproof · cited by 3,086
- le_antisymmproof · cited by 2,068
- Finset.finite_toSetproof · cited by 210
- Subgroup.closureproof · cited by 196
- Submonoid.closureproof · cited by 167
- Subgroup.toSubmonoidstatement and proof · cited by 114
- Set.Finite.unionproof · cited by 74
Cited by5
Results whose statement or proof uses this declaration.
- Group.fg_iff_monoid_fgproof · cited by 2
- Group.fg_iff_subgroup_fgproof · cited by 2
- AddSubgroup.fg_iff_mul_fgproof · cited by 1
- Subgroup.fg_iff_add_fgproof · cited by 1
- Subgroup.FG.prodproof · cited by 0