[Merged by Bors] - feat(GroupTheory/IndexNormal): subgroups of small index are normal#21186
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[Merged by Bors] - feat(GroupTheory/IndexNormal): subgroups of small index are normal#21186AntoineChambert-Loir wants to merge 15 commits intomasterfrom
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PR summary 2f870c8233Import changes exceeding 2%
|
| File | Base Count | Head Count | Change |
|---|---|---|---|
| Mathlib.Data.Nat.Prime.Factorial | 297 | 385 | +88 (+29.63%) |
Import changes for all files
| Files | Import difference |
|---|---|
Mathlib.Data.Nat.Choose.Dvd |
63 |
Mathlib.Data.Nat.Prime.Factorial |
88 |
Mathlib.Data.Finite.Perm (new file) |
737 |
Mathlib.GroupTheory.IndexNormal (new file) |
840 |
Declarations diff
+ card_perm
+ coprime_factorial_iff
+ normal_of_index_eq_minFac_card
+ normal_of_index_eq_one
+ normal_of_index_eq_two
You can run this locally as follows
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>
## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>The doc-module for script/declarations_diff.sh contains some details about this script.
No changes to technical debt.
You can run this locally as
./scripts/technical-debt-metrics.sh pr_summary
- The
relativevalue is the weighted sum of the differences with weight given by the inverse of the current value of the statistic. - The
absolutevalue is therelativevalue divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).
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Here's a golf:
/-- A subgroup of a finite group whose index is the smallest prime factor is normal -/
theorem normal_of_index_eq_smallest_prime_factor
(hHp : H.index = (Nat.card G).minFac) :
H.Normal := by
by_cases hG0 : Nat.card G = 0
· rw [hG0, minFac_zero] at hHp
exact normal_of_index_eq_two hHp
by_cases hG1 : Nat.card G = 1
· rw [hG1, minFac_one] at hHp
exact normal_of_index_eq_one hHp
suffices H.normalCore.relindex H = 1 by
convert H.normalCore_normal
exact le_antisymm (relindex_eq_one.mp this) (normalCore_le H)
have : Finite G := finite_of_card_ne_zero hG0
have index_ne_zero : H.index ≠ 0 := index_ne_zero_of_finite
rw [← mul_left_inj' index_ne_zero, one_mul, relindex_mul_index H.normalCore_le]
have hp : Nat.Prime H.index := hHp ▸ minFac_prime hG1
have hp' {l : ℕ} (h1 : l.Prime) (h2 : l ∣ H.normalCore.index) : H.index ≤ l :=
hHp ▸ minFac_le_of_dvd h1.two_le (h2.trans H.normalCore.index_dvd_card)
have h : H.normalCore.index ∣ H.index ! := by
rw [normalCore_eq_ker, index_ker, index_eq_card, ← Nat.card_perm]
exact card_subgroup_dvd_card (toPermHom G (G ⧸ H)).range
exact dvd_antisymm (dvd_prime hp h hp') (index_dvd_of_le H.normalCore_le)
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alreadydone
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Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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Co-authored-by: Yaël Dillies <yael.dillies@gmail.com>
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Thanks!
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…21186) * `Subgroup.normal_of_index_eq_smallest_prime_factor`: in a finite group `G`, a subgroup of index equal to the smallest prime factor of `Nat.card G` is normal. * `Subgroup.normal_of_index_two`: in a group `G`, a subgroup of index 2 is normal (This does not require `G` to be finite.)
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* factorial-dvd-int: (143 commits) Apply suggestions from code review feat(Factorial): k! divides the product of any k successive integers feat(CategoryTheory): creation of finite limits (#21320) chore: update Mathlib dependencies 2025-02-01 (#21328) chore(GroupTheory/SpecificGroups/Alternating.lean): follow last minute review of JX (#21314) feat: `‖x‖ₑ.toNNReal = ‖x‖₊` (#21306) chore: cleanup imports in Archive/IfNormalization (#21318) doc: fix several typos (#21315) feat(CategoryTheory): transfer being iso along an iso in the arrow category (#21310) chore: delete declarations deprecated between 2024-01 and 2024-07 (#21271) feat(Analysis/Normed/Module/Dual): polar in a normed space as a submodule (#20084) chore(Data/ZMod/Basic): split `ZMod.valMinAbs` off (#21308) feat(GroupTheory/Perm/Centralizer): study the centralizer of a permutation (#17522) feat(RingTheory/LocalRing): `IsLocalRing` for subrings (#21168) chore: update Mathlib dependencies 2025-02-01 (#21312) chore: update Mathlib dependencies 2025-01-31 (#21311) feat: generalize `mem_dite` to `Membership α β` (#21262) feat: Lemmas for some monomial orders (#16177) feat(CategoryTheory): the localized category is monoidal (#12728) feat: add function log⁺ (=positive part of the logarithm) and prove standard estimates (#21289) feat(RingTheory/WittVector): ring of Witt vectors is p-adically complete (#21295) feat(GroupTheory/GroupAction/Blocks): more on blocks (#21284) fix(FieldTheory/KrullTopology): make `krullTopology_discreteTopology_of_finiteDimensional` universe polymorphic (#21299) feat(RingTheory/Artinian): integral non-zero-divisors are units over artinian rings (#21199) refactor(Topology/Gluing): simplify definition of `TopCat.GlueData.Rel` (#20653) feat(RingTheory/PowerSeries): binomial series (#20192) chore(Mathlib/RingTheory/MvPolynomial): rename MonomiaOrder.lCoeff to MonomialOrder.leadingCoeff (#21290) chore (RingTheory/HahnSeries): fix names that use coeff (#21279) feat: let `notation3` distinguish `Prop` vs `Type _ ` vs `Sort _` (#21233) chore(MeasureTheory/Function/StronglyMeasurable): split Basic into Basic and AEStronglyMeasurable (#21273) feat(CategoryTheory): the monoidal category structure on a localization (#20951) feat(Analysis/Complex/Hadamard): generalize Hadamard's three lines theorem (#15009) feat(Order/CompleteBooleanAlgebra): Himp in terms of sSup (#20328) feat(ENNReal/Basic): add `ofNat_ne_top` and `top_ne_ofNat` (#14486) feat: Function.const as a PartialEquiv (#21137) chore(NonZeroDivisors): don't import rings (#20871) feat(Data/Set/Lattice): insert distributivity with iUnion/iInter (#21267) feat(GroupTheory/SpecificGroups/AlternatingGroup): subgroups of index 2 of Equiv.Perm (#21190) feat(GroupTheory/GroupAction/Transitive): basic results on transitive actions (#21285) perf(MeasureTheory/Function/LpSpace.lean): speed up (#21179) feat(Order): order isomorphisms from `Fin n` for small `n` (#21120) refactor(Topology/Group): turn morphisms in ProfiniteGrp into one field structures (#20740) feat: Sylow's first theorem for elementary `p`-groups (#21072) chore(Submonoid/Membership): don't import `MonoidWithZero` (#20748) refactor(Algebra/Algebra/Pi): cleanup and renaming (#21213) feat(GroupTheory/IndexNormal): subgroups of small index are normal (#21186) feat(Algebra/Group/Action): add definition of equidecomposition (#16936) feat(CategoryTheory/Subpresheaf): equalizer (#21096) feat: add lemmas about products of `Matrix.stdBasisMatrix` (#21204) chore: update Mathlib dependencies 2025-01-31 (#21282) ...
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…21186) * `Subgroup.normal_of_index_eq_smallest_prime_factor`: in a finite group `G`, a subgroup of index equal to the smallest prime factor of `Nat.card G` is normal. * `Subgroup.normal_of_index_two`: in a group `G`, a subgroup of index 2 is normal (This does not require `G` to be finite.)
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Subgroup.normal_of_index_eq_smallest_prime_factor: in a finite groupG,a subgroup of index equal to the smallest prime factor of
Nat.card Gis normal.Subgroup.normal_of_index_two: in a groupG, a subgroup of index 2 is normal(This does not require
Gto be finite.)