Residually finite group

A group $G$ is residually finite when every nonidentity $g\in G$ survives in some finite quotient: there are a finite group $Q$ and a homomorphism $\varphi :G\to Q$ with $\varphi (g)\ne 1$.

The finite residual consists of all elements killed by every homomorphism to a finite group. A group is a residually finite group precisely when its finite residual is trivial. Intersecting all finite-index subgroups gives the same result: the normal core of each such subgroup is normal of finite index and lies inside it.

Let $G$ be a finitely generated group and $\theta :G\to G$ a surjective endomorphism. Pullback by $\theta$ permutes the finite set of normal subgroups of any prescribed finite index. For each such $N$, some iterate satisfies ${\theta }^{-m}(N)=N$, so $\ker \theta \subseteq \ker {\theta }^m\subseteq N$. Intersecting proves the inclusion. Consequently every finitely generated residually finite group is a Hopfian group.

If $K$ is a finitely generated group, then the semidirect product $K⋊H$ is a residually finite group exactly when both $K$ and $H$ are residually finite groups. To separate an element whose $H$ coordinate is nontrivial, use a finite quotient of $H$. For $1\ne k\in K$, choose a finite-index normal subgroup $U$ omitting $k$, and intersect all subgroups of index at most $[K:U]$ to obtain a characteristic finite-index subgroup $C\le U$. The induced map into $(K/C)⋊\mathrm{im}(H\to \mathrm{Aut}(K/C))$ has finite target and separates $k$. More generally, finite generation can be replaced by a separating family of finite-index normal subgroups invariant under the given $H$ action.

A group $G$ is conjugacy separable when any two nonconjugate elements remain nonconjugate in some finite quotient. Equivalently, every conjugacy class is closed in the profinite topology.

Every subgroup of a residually finite group is residually finite, because a finite quotient separating an element of the ambient group also separates it after restriction to the subgroup.

For a prime number $p$, reducing every entry modulo $p$ defines a group homomorphism from an integral matrix group to the corresponding matrix group over ${\mathbb{F}}_p$. The target is finite. If an integral matrix $M$ is not the identity, choosing $p$ not to divide one nonzero entry of $M-I$ ensures that its reduction is also nonidentity.

An infinite residually finite group is not a simple group. For a nonidentity element, residual finiteness supplies a map to a finite group in which it survives. Its kernel is proper and cannot be trivial, since an infinite group cannot inject into a finite group.