TITLE: Infinite groups which contain all finite groups as subgroups
QUESTION [16 upvotes]: There are infinite graphs which contain all finite graphs as induced subgraphs, e.g. the Rado graph or the coprimeness graph on the naturals.

Are there infinite groups which
  contain all finite groups as
  subgroups?

REPLY [10 votes]: My favorite is Thompson's group $V$.  
My favorite picture of $V$ is to take a set $X$ which is a disjoint union of subsets $L$ and $R$ having fixed bijections $l:X\rightarrow L$ and $r:X\rightarrow R$.  Finite words in $l$ and $r$ map $X$ to "fragments" of $X$.  Two "fragments" $W$ and $U$ that are the images, respectively, of words $w$ and $u$ in $l$ and $r$ are connected by the bijection $uw^{-1}:W\rightarrow U$.  The two "fragments" will be disjoint iff neither of $w$ nor $u$ is a prefix of the other.  If this condition holds, let $(w,u)$ represent the permutation on $X$ that is $uw^{-1}$ on $W$, is $wu^{-1}$ on $U$, and is the identity elsewhere.  The group $V$ is generated by all such $(w,u)$.  It is finitely presented and contains all finite groups.
Other f.p. groups containing all finite groups are known as Houghton groups.  See the section on Houghton groups K. S. Brown "Finiteness properties of groups" in Journal of Pure and Applied Algebra, 44 (1987), 45-75.  I forget how they are indexed, but from about n=3 on up, they are all f.p.

REPLY [2 votes]: I can't resist mentioning that the group of unitary elements in the hyperfinite $II_{1}$ factor (The group von Neumann algebra of Bruce Westbury's "locally finite" permutation group above) in fact contains every countable discrete amenable group as a subgroup.