feat(GroupTheory/PCore): define the p-core of a subgroup

[kim-em]

This PR adds Subgroup.pCore p H, the largest normal p-subgroup of a subgroup H : Subgroup G (classically O_p(H)), defined as the supremum of all normal p-subgroups. Even without finiteness or primality hypotheses, this is itself a normal p-group (the family of normal p-subgroups is directed under ≤, so every element of the supremum lies in some specific summand). For the classical O_p(G), take pCore p (⊤ : Subgroup G).

API covers the universal-property characterisations (le_pCore, normal_le_pCore, mem_pCore_iff, pCore_eq_bot_iff, pCore_eq_top_iff, pCore_zero, pCore_one), the Sylow alignment pCore p H = ⨅ P : Sylow p H, (P : Subgroup H), and behaviour under group homomorphisms via MonoidHom.subgroupMap and MonoidHom.subgroupComap (map_pCore_le_pCore, comap_pCore_eq_pCore and friends, MulEquiv.map_pCore).

Also adds Subgroup.normal_iSup_normal in Mathlib/Algebra/Group/Subgroup/Pointwise.lean (the supremum of a family of normal subgroups is normal) as the sibling of the existing normal_iInf_normal, and ker_subgroupComap in Mathlib/Algebra/Group/Subgroup/Ker.lean alongside the existing ker_subgroupMap. Both marked @[to_additive].

Used downstream in leanprover/lean-eval#311 to state the Baer-Suzuki theorem as a lean-eval problem.

🤖 Prepared with Claude Code

[github-actions]

PR summary b1fc61ea28

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.GroupTheory.PCore (new file)	1176

---

Declarations diff

+ _root_.MulEquiv.map_pCore
+ comap_pCore_eq_pCore
+ comap_pCore_le_pCore
+ directed_pCore
+ instance : Nonempty {N : Subgroup H // N.Normal ∧ IsPGroup p N}
+ isPGroup_iInf_sylow
+ isPGroup_pCore
+ isPGroup_pCore_subgroupOf
+ ker_subgroupComap
+ le_pCore
+ le_pCore_of_le
+ le_pCore_subgroupOf
+ le_pCore_subgroupOf_iff
+ map_pCore_eq_pCore
+ map_pCore_eq_pCore_of_ker_isPGroup
+ map_pCore_le_pCore
+ map_subtype_le_pCore_iff
+ mem_pCore_iff
+ mem_pCore_subgroupOf_iff
+ normal_iInf_sylow
+ pCore
+ pCore_bot
+ pCore_eq_bot_iff
+ pCore_eq_iInf_sylow
+ pCore_eq_iSup
+ pCore_eq_self
+ pCore_eq_self_iff
+ pCore_le
+ pCore_le_comap_pCore
+ pCore_le_sylow
+ pCore_one
+ pCore_subgroupOf
+ pCore_subgroupOf_characteristic
+ pCore_subgroupOf_eq_iInf_sylow
+ pCore_subgroupOf_eq_sylow_iff_normal
+ pCore_subgroupOf_le_sylow
+ pCore_subgroupOf_normal
+ pCore_zero
+ subgroupOf_comap_subgroupComap
+ subgroupOf_map_subgroupMap
+ sylow_eq_pCore_subgroupOf_of_normal
+ top_map_subtype

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit b1fc61ea28
Reference commit 6da0491965

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[tb65536]

Oh, great! This might be relevant for some stuff @Whysoserioushah has been thinking about.

[SnirBroshi]

Hello! I just found this PR when I went to open my own PR defining p-cores 😅

I'll try to review this soon to see if I find stuff worth keeping from my take on this, and in the meantime I opened it as a draft PR if people are interested to play spot-the-difference until I get to it: #40036
(I'll close it after I suggest stuff here)

[kim-em]

@tb65536, I see you marked this back as awaiting-author, but I only see a conversation between you and @SnirBroshi, which I think concludes with agreeing with the status quo?

I'm going to remove awaiting-author again, but please point me to anything I've missed.

[SnirBroshi]

The status quo is

def pCore (p : ℕ) (H : Subgroup G) : Subgroup H :=
  ⨆ N : {N : Subgroup H // N.Normal ∧ IsPGroup p N}, (N : Subgroup H)

but we've suggested

def pCore (p : ℕ) (H : Subgroup G) : Subgroup G :=
  ⨆ N ≤ H, ⨆ _ : (N.subgroupOf H).Normal, ⨆ _ : IsPGroup p N, N

Specifically the issue is that pCore p changes the ambient group, but should be Subgroup G → Subgroup G.

[SnirBroshi]

This seems really awkward to me, could you use

⨆ N ≤ H, ⨆ _ : (N.subgroupOf H).Normal, ⨆ _ : IsPGroup p N, N

instead?
If it makes the proofs worse for some reason, then could you instead add versions of pCore_eq_iSup and mem_pCore_iff with this def?

[rosborn]

I'm not sure if this should be a Zulip post, so let me know. I've been refamiliarizing myself with mathlib's group theory api and have noticed IsPGroup, (recently) primaryComponent, and now pCore are all prime agnostic (for composite n, they only care about n.primeFactors). pCore n H at composite n is already O_{primes n}(H), primaryComponent G n is the π-primary component, etc. Should we make this clearer in the docstrings? As far as I know, this deviates from the standard literature, but in my own experiments, parameterizing by n is easier to work with than translating to a set of primes.

[rosborn]

directed_pCore can be deleted with the suggested changes to mem_pCore_subgroupOf_iff / mem_pCore_iff / pCore_eq_bot_iff.

normal_iInf_sylow / isPGroup_iInf_sylow no longer have any pCore-specific neighbors. Might be worth moving to Sylow.lean?

[rosborn]

This is a corollary of map_pCore_eq_pCore_of_ker_isPGroup

map_pCore_eq_pCore_of_ker_isPGroup (e : G →* G') (by
  rw [(e : G →* G').ker_eq_bot e.injective]
  exact IsPGroup.of_bot)

[rosborn]

Also requires import Mathlib.Algebra.Group.Subgroup.Ker

[rosborn]

Suggested change

	rw [pCore, ← Subgroup.map_iSup]
	haveI : ∀ N : {N : Subgroup H // N.Normal ∧ IsPGroup p N}, ((N : Subgroup H)).Normal :=
	fun N => N.2.1
	exact (Sylow.iSup_of_normal _ fun N => N.2.2).map H.subtype
	isPGroup_pCore_subgroupOf.of_equiv (subgroupOfEquivOfLe pCore_le)

[rosborn]

Can also add an instance for normality

private instance (N : {N : Subgroup H // N.Normal ∧ IsPGroup p N}) : (N : Subgroup H).Normal :=
  N.2.1

Will get rid of the haveIs in isPGroup_pCore_subgroupOf and pCore_subgroupOf_normal.

[rosborn]

Is there a reason these are flipped?

[rosborn]

Suggested change

	x ∈ (pCore p H).subgroupOf H ↔ ∃ N : Subgroup H, N.Normal ∧ IsPGroup p N ∧ x ∈ N := by
	rw [pCore_subgroupOf, mem_iSup_of_directed directed_pCore]
	exact ⟨fun ⟨N, hxN⟩ => ⟨N, N.2.1, N.2.2, hxN⟩,
	fun ⟨N, hN, hP, hxN⟩ => ⟨⟨N, hN, hP⟩, hxN⟩⟩
	x ∈ (pCore p H).subgroupOf H ↔ ∃ N : Subgroup H, N.Normal ∧ IsPGroup p N ∧ x ∈ N :=
	⟨fun hx => ⟨(pCore p H).subgroupOf H, pCore_subgroupOf_normal, isPGroup_pCore_subgroupOf, hx⟩,
	fun ⟨_, hN, hP, hxN⟩ => le_pCore_subgroupOf hN hP hxN⟩

[rosborn]

Suggested change

	x ∈ pCore p H ↔ ∃ N : Subgroup H, N.Normal ∧ IsPGroup p N ∧ x ∈ N.map H.subtype := by
	rw [pCore_eq_iSup, mem_iSup_of_directed fun a b =>
	(directed_pCore a b).imp fun _ h => ⟨map_mono h.1, map_mono h.2⟩]
	exact ⟨fun ⟨N, hxN⟩ => ⟨N, N.2.1, N.2.2, hxN⟩,
	fun ⟨N, hN, hP, hxN⟩ => ⟨⟨N, hN, hP⟩, hxN⟩⟩
	x ∈ pCore p H ↔ ∃ N : Subgroup H, N.Normal ∧ IsPGroup p N ∧ x ∈ N.map H.subtype :=
	⟨fun hx => ⟨(pCore p H).subgroupOf H, pCore_subgroupOf_normal, isPGroup_pCore_subgroupOf,
	by rwa [map_subgroupOf_eq_of_le pCore_le]⟩,
	fun ⟨_, hN, hP, hxN⟩ => le_pCore hN hP hxN⟩

[rosborn]

Suggested change

	have key : pCore p H = ⊥ ↔ (pCore p H).subgroupOf H = ⊥ :=
	⟨fun h => by rw [h, bot_subgroupOf],
	fun h => by rw [← map_subgroupOf_eq_of_le (pCore_le (H := H)), h, map_bot]⟩
	rw [key, pCore_subgroupOf]
	refine ⟨fun h N hN hP => le_bot_iff.mp (h ▸ le_iSup
	(fun N : {N : Subgroup H // N.Normal ∧ IsPGroup p N} => (N : Subgroup H)) ⟨N, hN, hP⟩),
	fun h => ?_⟩
	rw [eq_bot_iff_forall]
	intro x hx
	obtain ⟨N, hxN⟩ := (mem_iSup_of_directed directed_pCore).mp hx
	simpa [h N N.2.1 N.2.2] using hxN
	refine ⟨fun h N hN hP => ?_, fun h => ?_⟩
	· have hle := le_pCore hN hP
	rwa [h, le_bot_iff, map_eq_bot_iff_of_injective _ H.subtype_injective] at hle
	· rw [← map_subgroupOf_eq_of_le (pCore_le (H := H)),
	h _ pCore_subgroupOf_normal isPGroup_pCore_subgroupOf, Subgroup.map_bot]

[rosborn]

Since #40012 merged, this can be shortened.

Suggested change

	theorem pCore_subgroupOf_le_sylow (P : Sylow p H) : (pCore p H).subgroupOf H ≤ P := by
	have hpsup : IsPGroup p ((P : Subgroup H) ⊔ (pCore p H).subgroupOf H : Subgroup H) :=
	P.2.to_sup_of_normal_right isPGroup_pCore_subgroupOf
	have heq : (P : Subgroup H) ⊔ (pCore p H).subgroupOf H = P := P.3 hpsup le_sup_left
	exact le_sup_right.trans heq.le
	theorem pCore_subgroupOf_le_sylow (P : Sylow p H) : (pCore p H).subgroupOf H ≤ P :=
	isPGroup_pCore_subgroupOf.le_sylow_of_normal P