# Groups which satisfy Mal'cev's theorem (locally residually finite)

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?

-
When one has a property of groups, and it holds for finitely-generated subgroups of a group, then the adjective "locally" is usually pinned on. So I would call your property "locally residually finite". –  Ian Agol Jul 9 '11 at 22:03
Ah, thanks. That's useful. –  Benjamin Antieau Jul 9 '11 at 22:36
Stallings proves in his lecture notes that GL_n(R) is locally residually finite for any commutative ring R with unit. See math.berkeley.edu/~stall/math257 –  Benjamin Steinberg Jul 20 '11 at 14:38
Yes. This statement is Mal'cev's theorem. Thanks for the link to the notes though. Those look nice. –  Benjamin Antieau Jul 20 '11 at 18:31
I think Malcev only proved this for the case of a field. The ring case is harder, –  Benjamin Steinberg Jul 30 '11 at 12:20

If $G$ is a compact group, it has this property. This is because by Peter-Weyl compact groups are residually linear. Now use Malcev.

-
Good example, although perhaps I should have said I wanted non-residually linear groups as well. –  Benjamin Antieau Jul 9 '11 at 22:35
So far, all listed examples are direct limits of residually linear groups. What are completely different examples? –  Benjamin Steinberg Jul 10 '11 at 2:35

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

-
Indeed locally finite groups, by definition, are locally residually finite. It is known that, for every prime $p$, there are uncountably many pairwise non-isomorphic locally finite groups which moreover are $p$-groups: see Burns, R. G. A wreath tower construction of countably infinite, locally finite groups. Math. Z. 105, 1968, 367–386 –  Alain Valette Jul 10 '11 at 20:52
This is a subgroup of my example. –  Ian Agol Jul 10 '11 at 21:00
There are simple examples, but probably not what you're looking for, such as a union $GL_{\infty}(k)=\cup_n GL_n(k)$, where $GL_n(k) \subset GL_{n+1}(k)$ embeds in the obvious way by thinking of $k^{n+1}=k^n \oplus k$.