Isomorphic congruence groups and Hecke operators

Extract

Let G, H, K be groups such that G is normal in K and G ⊆ H ⊆ K. Let I(H, K) be the set of inner automorphisms of K restricted to H; thus α ∊ I(H, K) if and only if, for some κ∊ K, α(h) = k^-1hk for all h ∊ H. Let φ be an isomorphism of H/G onto a subgroup H^φ/G of K/G. An isomorphism Φ of H onto H^(φ) is called an extension of ø if

Φ(h)G = φ(hG) for all h∊H.