Theorems · Theorem · group theory
AddSubgroup.IsSubnormal.isSubnormal_iff
∀ {G : Type u_1} [inst : AddGroup G] {H : AddSubgroup G},
H.IsSubnormal ↔ ∃ n f, Monotone f ∧ (∀ (i : ℕ), ((f i).addSubgroupOf (f (i + 1))).Normal) ∧ f 0 = H ∧ f n = ⊤A characterisation of satisfying IsSubnormal in terms of chains of additive
subgroups, each normal in the following one.
The sequence stabilises once it reaches ⊤, which is guaranteed at the asserted n.
- Defined in
- Mathlib.GroupTheory.IsSubnormal
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddGroup
Around this declaration
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Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Top.topstatement and proof · cited by 9,680
- AddGroupstatement and proof · cited by 4,410
- AddSubgroupstatement and proof · cited by 3,232
- Monotonestatement and proof · cited by 1,397
- le_topproof · cited by 411
- AddSubgroup.Normalstatement and proof · cited by 183
- AddSubgroup.addSubgroupOfstatement and proof · cited by 87
- monotone_nat_of_le_succproof · cited by 45
- AddSubgroup.IsSubnormalstatement and proof · cited by 17
- AddSubgroup.addSubgroupOf_selfproof · cited by 3
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