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?
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Sign up to join this communityThere 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?
Yes, plenty. The group only has to contain all finite permutation groups. Perhaps the most straightforward example would be the permutations of a countable set. That is bijections which fix all but a finite set.
There is a finitely presented group which contains all finitely presented groups, as proved by Higman.
There many examples for instance the direct product of all finite groups. A more interesting example of a finitely generated (topologically) profinite groups is $\Pi_{n \geq 5} A_n$. Of course you can also construct many examples using direct limits of finite groups.
Another interesting example is the Nottingham group which is for $p>3$ a finitely presented pro-$p$ group (proved by Mikhail Ershov) which contains every countably based pro-$p$ group (proved by Rachel Camina and aslo by Ivan Fesenko) and in particuar every finite $p$-group. Of course it does not contain every finite group, but still in the spirit of your question. The Nottingham group has many other remarkable properties and is a worth knowing example of a pro-$p$ group.
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.
Here's a finitely generated group that contains all finite groups: permutations of the integers that are almost shifts; that is, each permutation differs form the shift-by-$n$ permutation by a permutation with finite support. This maps to the integers with kernel Bruce's group. Is it finitely presented?
$\{[s_7,s_{7+k}]|k\in\mathbb{Z}\}$
.
$\endgroup$
Jul 22, 2010 at 0:11
Another interesting example is Hall's universal group
http://en.wikipedia.org/wiki/Hall%27s_universal_group
Hall's universal group is a countable locally finite group, say U, which is uniquely characterized by the following properties.
* Every finite group G admits a monomorphism to U.
* All such monomorphisms are conjugate by inner automorphisms of U.
If you want your group to be countable, you may consider the direct sum of all S_n's.
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.