# Thread: residually finite groups

1. ## residually finite groups

I have 2 definitions for residually finite groups.

Definition 1: $G$ is said to be residually finite if for each nontrivial element, $x$ in $G$, there exists a finite index normal subgroup $N_x$ in G such that $x \notin N_x$.

Definition 2: $G$ is said to be residually finite if for each nontrivial element, $x \in G$, there exists a homomorphism $f$ from $G$ 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 $1 \longrightarrow K \longrightarrow G \longrightarrow Q \longrightarrow 1$, with $K$ is residually finite and $Q$ is finite.

Can anyone show me where to start or some hints?

2. can anyone give me some hints or show me how to start?