TITLE: Groups which satisfy Mal'cev's theorem (locally residually finite)
QUESTION [12 upvotes]: Recall that a group $G$ is residualy finite if for every non-zero element $g\in G$ there exists a homomorphism $\sigma:G\rightarrow H$ such that $H$ is finite and $\sigma(g)\neq 0$. Mal'cev's theorem says that if $k$ is a field then any finitely-generated subgroup of $GL_n(k)$ is residually finite. For a proof of this theorem, see Steve D (Smith?)s answer to this question MO:9628. Note that the theorem does not say that $GL_n(k)$ is residually finite.

Does anyone know any other classes of groups with the property that any finitely generated subgroup is residually finite.

I would prefer examples with the following two properties: first, the group is not itself residually finite, and second, it is not simply a subgroup of some $GL_n(k)$. So, this excludes, for instance, free groups.
Side question: is there a good name for this property?

REPLY [4 votes]: There are uncountably many Tarski monsters, which are finitely generated simple infinity groups whose proper subgroups are cyclic. No infinite simple group is residually finite, so Tarski monsters aren't, but their proper subgroups obviously are residually finite.

REPLY [3 votes]: Perhaps a locally finite simple group would help?  How about the subset of all even permutations of the natural numbers with finite support?  Unless I am misremembering something, this should be a simple locally finite group that is not a subgroup of GL_n(k).
Gerhard "Email Me About System Design" Paseman, 2011.07.09