An injective homomorphism ϕ ϕ : G ′ → → G factors canonically as an isomorphism followed by an inclusion, G ′ → → ∼ ∼ H ↪ ↪ G for some subgroup ⁠ H ⁠ of ⁠ G ⁠. Injective homomorphisms are the monomorphisms in the category of groups.