For example, the dihedral group D 4 is generated by the right rotation r 1 and the reflection f v in a vertical line (every element of D 4 is a finite product of copies of these and their inverses). Hence there is a surjective homomorphism ⁠ ϕ ϕ ⁠ from the free group ⟨ ⟨ r , f ⟩ ⟩ on two generators to D 4 sending r to r 1 and f to ⁠ f 1 ⁠. Elements in ker ⁡ ⁡ ϕ ϕ are called relations ; examples include ⁠ r 4 , f 2 , ( r ⋅ ⋅ f ) 2 ⁠. In fact, it turns out that ker ⁡ ⁡ ϕ ϕ is the smallest normal subgroup of ⟨ ⟨ r , f ⟩ ⟩ containing these three elements; in other words, all relations are consequences of these three. The quotient of the free group by this normal subgroup is denoted ⁠ ⟨ ⟨ r , f ∣ ∣ r 4 = f 2 = ( r ⋅ ⋅ f ) 2 = 1 ⟩ ⟩ ⁠. This is called a presentation of D 4 by generators and relations, because the first isomorphism theorem for ⁠ ϕ ϕ ⁠ yields an isomorphism ⁠ ⟨ ⟨ r , f ∣ ∣ r 4 = f 2 = ( r ⋅ ⋅ f ) 2 = 1 ⟩ ⟩ → → D 4 ⁠.