feat(GroupTheory/Commutator/Basic): ⁅H₁, H₂⁆ is a normal subgroup of H₁ ⊔ H₂#39227
Open
SnirBroshi wants to merge 1 commit into
Open
Conversation
PR summary 686c76b39dImport changes for modified filesNo significant changes to the import graph Import changes for all files
|
tb65536
reviewed
May 13, 2026
Comment on lines
+196
to
+206
| @[to_additive] | ||
| instance normal_subgroupOf_commutator_sup : (⁅H₁, H₂⁆.subgroupOf <| H₁ ⊔ H₂).Normal := by | ||
| refine normal_subgroupOf_of_le_normalizer <| sup_le ?_ ?_ <;> | ||
| apply le_normalizer_closure_iff.mpr <;> | ||
| rintro g hg _ ⟨g₁, hg₁, g₂, hg₂, rfl⟩ | ||
| · apply (mul_mem_cancel_right <| commutator_mem_commutator hg hg₂).mp | ||
| rw [← commutatorElement_mul_left_eq_conj_mul g g₁ g₂] | ||
| exact commutator_mem_commutator (mul_mem hg hg₁) hg₂ | ||
| · apply (mul_mem_cancel_left <| commutator_mem_commutator hg₁ hg).mp | ||
| rw [← mul_assoc, ← mul_assoc, ← commutatorElement_mul_right_eq_mul_conj] | ||
| exact commutator_mem_commutator hg₁ <| mul_mem hg hg₂ |
Contributor
There was a problem hiding this comment.
Would you mind splitting up the proof like this:
@[to_additive]
instance normalizer_commutator_ge_left : H₁ ≤ normalizer (⁅H₁, H₂⁆ : Subgroup G) := by
sorry
@[to_additive]
instance normalizer_commutator_ge_right : H₂ ≤ normalizer (⁅H₁, H₂⁆ : Subgroup G) := by
rw [commutator_comm]
apply normalizer_commutator_ge_left
@[to_additive]
instance normal_subgroupOf_commutator_sup' : (⁅H₁, H₂⁆.subgroupOf <| H₁ ⊔ H₂).Normal :=
normal_subgroupOf_of_le_normalizer <| sup_le
(normalizer_commutator_ge_left H₁ H₂) (normalizer_commutator_ge_right H₁ H₂)
tb65536
added
the
awaiting-author
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
2 participants
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
Shows
⁅H₁, H₂⁆ ≤ H₁ ⊔ H₂and(⁅H₁, H₂⁆.subgroupOf <| H₁ ⊔ H₂).Normal, and adds_ ≤ normalizer _ ↔lemmas to help.(from the book "Finite Groups" by Daniel Gorenstein)
Also renames
AddSubgroup.mem_normalizer_iff_conj_image_eqbecauseto_additivenow renamesconjtoaddConjand this theorem should stay in sync with its counterpart, even though it's not usingto_additivebecauseMulAutisn't additivized.The proofs seem pretty long for such trivial lemmas, suggestions welcome :)