Theorems · Theorem · group theory
Group.isCyclic_prod_iff
∀ {M : Type u_4} {N : Type u_5} [inst : Group M] [inst_1 : Group N],
IsCyclic (M × N) ↔ IsCyclic M ∧ IsCyclic N ∧ (Nat.card M).Coprime (Nat.card N)The product of two finite groups is cyclic iff both of them are cyclic and their orders are coprime.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites34
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Top.topproof · cited by 9,680
- Groupstatement and proof · cited by 6,238
- Bot.botproof · cited by 4,720
- MonoidHomproof · cited by 3,629
- Subgroupproof · cited by 3,593
- Finiteproof · cited by 3,029
- Nontrivialproof · cited by 2,416
- MulEquivproof · cited by 1,142
- CommGroupproof · cited by 990
- Nat.cardstatement and proof · cited by 844
- Infiniteproof · cited by 352
- MonoidHom.kerproof · cited by 212
Cited by2
Results whose statement or proof uses this declaration.
- ZMod.isCyclic_units_four_mul_iffproof · cited by 1
- ZMod.not_isCyclic_units_of_mul_coprimeproof · cited by 1