residually finite groups

**deniselim17** · February 17th 2009, 11:52 PM

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?

**deniselim17** · February 18th 2009, 06:05 PM

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

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