I have 2 definitions for residually finite groups.
Definition 1: is said to be residually finite if for each nontrivial element, in , there exists a finite index normal subgroup in G such that .
Definition 2: is said to be residually finite if for each nontrivial element, , there exists a homomorphism from to a finite group, such that f(x) is not the identity.
I have a problem in proving finite extension of residually finite groups are residually finite.
I have , with is residually finite and is finite.
Can anyone show me where to start or some hints?