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Problem
Let $G=N \rtimes_{\phi} Q$ where $N\lhd G$ finitely generated and residually finite and $Q$ residually finite. Show that $G$ is residually finite
Attempt
I know that
$N$ residually finite and f.g. implies $N$ is Hopfian
$N$ residually finite and f.g. implies $Aut(N)$ is residually fin...
In the mathematical field of group theory, a group G is residually finite or finitely approximable if for every element g that is not the identity in G there is a homomorphism h from G to a finite group, such that
h
(
g
)
≠
1.
{\displaystyle h(g)\neq 1.\,}
There are a number of equivalent definitions:
A group is residually finite if for each non-identity element in the group, there is a normal subgroup of finite index not containing that element.
A group is residually finite if and only if the intersection...
