feat(GroupTheory): coatoms of the subgroup lattice#41651
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Add three group-theory results: * `QuotientGroup.comapMk'OrderIso'`: the correspondence theorem as an order isomorphism onto `Set.Ici N` (interval-order variant of `comapMk'OrderIso`). * `isCyclic_of_isCoatom_subsingleton`: a group with at most one maximal subgroup is cyclic (only `IsCoatomic (Subgroup G)` needed). * `CommGroup.isSimpleGroup_iff_isCoatom`: a subgroup of a commutative group is maximal iff the quotient by it is simple (group analogue of `isSimpleModule_iff_isCoatom`).
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PR summary 9057cb64ddImport changes for modified filesNo significant changes to the import graph Import changes for all files
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| /-- A group with at most one maximal subgroup is cyclic. Maximal subgroups are the coatoms | ||
| `IsCoatom (· : Subgroup G)` of the subgroup lattice; only `IsCoatomic (Subgroup G)` is required | ||
| (automatic for finite `G`), with no finiteness, commutativity, or `p`-group hypothesis. -/ | ||
| @[to_additive /-- An additive group with at most one maximal subgroup is cyclic. Maximal subgroups | ||
| are the coatoms `IsCoatom (· : AddSubgroup G)` of the subgroup lattice; only | ||
| `IsCoatomic (AddSubgroup G)` is required (automatic for finite `G`), with no finiteness, | ||
| commutativity, or `p`-group hypothesis. -/] |
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Can we not mention what's not a hypothesis and instead only talk about what is a hypothesis?
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Add three group-theory results related to maximal subgroups:
isCyclic_of_isCoatom_subsingleton: a group with at most one maximal subgroup is cyclic.CommGroup.isSimpleGroup_iff_isCoatom: a subgroup of a commutative group is maximal iff the quotient by it is simple (group analogue ofisSimpleModule_iff_isCoatom).QuotientGroup.comapMk'OrderIso': the correspondence theorem as an order isomorphism onto the intervalSet.Ici N(interval-order variant ofcomapMk'OrderIso, whose codomain is the bare subtype), making theSet.Iciorder API available, mirroringSubmodule.comapMkQRelIso.🤖 This PR was extracted from the SKW project by Claude.