The first isomorphism theorem produces the desired isomorphism, which sends the coset v + M T R ℓ ℓ to K T v . This corresponds to rewriting the tuple ( log ⁡ ⁡ q 1 , log ⁡ ⁡ q 2 , … … , log ⁡ ⁡ q n ) into the pi groups ( log ⁡ ⁡ π π 1 , log ⁡ ⁡ π π 2 , … … , log ⁡ ⁡ π π p ) coming from the columns of K .