Group Theory – Is There a Universal Countable Group?

gr.group-theory

This recent MO
question,
answered now several times over, inquired whether an
infinite group can contain every finite group as a
subgroup. The answer is yes by a variety of means.

So let us raise the stakes: Is there a countable group
containing (a copy of) every countable group as a subgroup?

The countable random graph, after all, which inspired the
original question, contains copies of all countable graphs,
not merely all finite graphs. Is this possible with groups?
What seems to be needed is a highly saturated countable group.

  • An easier requirement would insist that
    the group contains merely all finitely generated groups as
    subgroups, or merely all countable abelian groups.
    (Reducing to a countable family, however, trivializes the question via the direct sum.)

  • A harder requirement would find the subgroups in particularly nice ways: as direct summands or as normal subgroups.

  • Another strengthened requirement would insist on an
    amalgamation property: whenever
    $H_0\lt H_1$ are finitely generated, then every copy of $H_0$ in the universal group $G$
    extends to a copy of $H_1$ in $G$. This property implies
    that $G$ is universal for all countable groups, by adding
    one generator at a time. This would generalize the
    saturation property of the random graph.

  • If there is a universal countable group, can one find a
    finitely generated such group, or a finitely presented
    such group? (This would lose amalgamation, of course.)

  • Moving higher, for which cardinals $\kappa$
    is there a universal group of size $\kappa$? That is, when is there
    a group of size
    $\kappa$ containing as a subgroup a copy of every group of size
    $\kappa$?

  • Moving lower, what is the minimum size of a finite group
    containing all groups of finite size at most $n$ as subgroups?
    Clearly, $n!$ suffices. Can one do better?

Best Answer

There isn't a countable group which contains a copy of every countable group as a subgroup. This follows from the fact that there are uncountably many 2-generator groups up to isomorphism.

The first example of such a family was discovered by B.H. Neumann. A clear account of his construction can be found in de la Harpe's book on geometric group theory.

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