let G be residually finite and H a subgroup of G. let then there exists a finite group and a homomorphism such that now consider

let be a family of residually finite groups and put let if then there exists a finite group and homomorphism such that2. Direct products of residually finite groups are residually finite.

if then define and note that since G is a direct product, for only finitely many thus is a finite group. define

then is a homomorphism and

i'm not sure exactly what you mean here! so we have an exact sequence of groups: now which one is assumed to be finite and which one residually finite?3. Finite extensions of residually finite groups are residually finite.

no! because every group is a quotient of a free group and all free groups are residually finite. so if the above was correct, then every group would be residually finite, which is obviously not true.are quotient groups of residually finite groups again residually finite? why? counter example?