Exercise 3, § 4

Recall that a group Γ is said to be residually finite if the intersection of the subgroups of finite index of Γ is reduced to $\{e\}$ (A, I, p. 129, exerc. 5, b)).

For a group Γ, let $\widehat{\Gamma}$ denote the projective limit $\widehat{\Gamma} =$ lim$\Gamma /H$, where H

runs through the set of normal subgroups of finite index of Γ, ordered by inclusion. One says that $\widehat{\Gamma}$ is the profinite completion of Γ.

a) A group Γ is residually finite if and only if, for every $g\in \Gamma$ different from the identity element, there exist a finite group F and a homomorphism $\varphi : \Gamma \rightarrow F$ such that $\varphi (\gamma )\not =e$.

b) Now prove that Γ is residually finite if and only if the canonical homomorphism from Γ into $\widehat{\Gamma}$ is injective.

c) Let K be a field. Every finitely generated subgroup (cf. A, I, p. 145, exerc. 16) of $\mathbf{G}\mathbf{L}(n,K)$ is residually finite ("Mal'tsev's theorem"). (Show that there exist an integer $m\geqslant 0$, a ring A equal to $\mathbf{Z}$ or to a finite field, and an injective homomorphism of such a subgroup into $\mathbf{G}\mathbf{L}(m,R)$, where R is of the form $A[X_1, . . . ,X_r,1/f].$)

d) Let Γ be a finitely generated discrete group. Suppose that there exists an injective homomorphism of Γ into a compact topological group. For every $\gamma \in \Gamma$ different from the identity element, there exists a finite-dimensional unitary irreducible representation of Γ such that the image of $\gamma$ under $\pi$ is not the identity mapping.

e) A finitely generated group is residually finite if and only if it is isomorphic to a subgroup of a compact group.