[Merged by Bors] - feat(group_theory/double_cosets): definition of double cosets and some basic lemmas. #9490
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| lemma subgroup_mul_singleton {H : subgroup G} {h : G} (hh : h ∈ H) : | ||
| (H : set G) * {h} = H := | ||
| begin | ||
| refine le_antisymm _ (λ h' hh', | ||
| ⟨h' * h⁻¹, h, H.mul_mem hh' (H.inv_mem hh), rfl, inv_mul_cancel_right h' h⟩), | ||
| rintros _ ⟨h', h, hh', rfl : _ = _, rfl⟩, | ||
| exact H.mul_mem hh' hh, | ||
| end | ||
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| lemma singleton_mul_subgroup {H : subgroup G} {h : G} (hh : h ∈ H) : | ||
| {h} * (H : set G) = H := | ||
| begin | ||
| refine le_antisymm _ (λ h' hh', ⟨h, h⁻¹ * h', rfl, H.mul_mem (H.inv_mem hh) hh', | ||
| mul_inv_cancel_left h h'⟩), | ||
| rintros _ ⟨h, h', rfl : _ = _, hh', rfl⟩, | ||
| exact H.mul_mem hh hh', | ||
| end |
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This should work on submonoids too, right?
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Does it? the proof uses inverses.
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Actually 2 * ℕ\{0} isnt ℕ\{0} so I dont think it works. unless the defn of monoid is not what I think it is.
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…e basic lemmas. (#9490) This contains the definition of double cosets and some basic lemmas about them, such as "the whole group is the disjoint union of its double cosets" and relationship to usual group quotients.
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This contains the definition of double cosets and some basic lemmas about them, such as "the whole group is the disjoint union of its double cosets" and relationship to usual group quotients.