Theorems · Theorem · group theory
Subgroup.card_subgroup_dvd_card
- #71 of the 100 theorems: Order of a Subgroup
- 1000+ list: Lagrange's theorem (group theory)
∀ {α : Type u_1} [inst : Group α] (s : Subgroup α), Nat.card ↥s ∣ Nat.card αLagrange's Theorem: The order of a subgroup divides the order of its ambient group. [Wikidata Q505798](https://www.wikidata.org/wiki/Q505798)
- Defined in
- Mathlib.GroupTheory.Coset.Card
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 95 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.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Groupstatement and proof · cited by 6,238
- Subgroupstatement and proof · cited by 3,593
- HasQuotient.Quotientproof · cited by 2,301
- Nat.cardstatement and proof · cited by 844
- CommSemigroupproof · cited by 62
- dvd_mul_leftproof · cited by 47
- Subgroup.card_eq_card_quotient_mul_card_subgroupproof · cited by 8
Cited by8
Results whose statement or proof uses this declaration.
- Subgroup.eq_bot_or_eq_top_of_prime_cardproof · cited by 3
- Subgroup.card_dvd_of_injectiveproof · cited by 3
- IsCyclic.normalizer_le_centralizerproof · cited by 2
- Sylow.card_eq_multiplicityproof · cited by 2
- NumberField.InfinitePlace.even_nat_card_aut_of_not_isUnramifiedproof · cited by 1
- isZGroup_of_coprimeproof · cited by 1
- IsZGroup.of_squarefreeproof · cited by 0
- Subgroup.normal_of_index_eq_minFac_cardproof · cited by 0