# Is there an infinite group that contains every finite group (and no infinite group) as a subgroup?

Question is in the title. For bonus points, construct the group $G$ such that it also has no infinite proper subgroups. (This second question relates to the Prüfer group, but that group is abelian, and clearly $G$ is nonabelian since it has nonabelian subgroups.)

Ignoring the second constraint for now, it is clear that the direct product of every finite group contains every finite group as a subgroup, but it is not a very "natural" group. Are there any examples of more common infinite groups that also happen to have every finite group as a subgroup?

• I am pretty sure this has been asked before: did you search? – Mariano Suárez-Álvarez Jan 7 '14 at 20:51
• @MarianoSuárez-Alvarez The second criterion, not in the title, is a bit different from most other questions. – Mario Carneiro Jan 7 '14 at 21:03
• Have you actually looked at the other questions? – Mariano Suárez-Álvarez Jan 7 '14 at 21:05
• @MarianoSuárez-Alvarez You don't seem to be giving links, so I will: Group where you can see all the finite subgroups. is similar, but does not consider the second constraint. – Mario Carneiro Jan 7 '14 at 21:09
• I am trying to get you to give the links, in fact. – Mariano Suárez-Álvarez Jan 7 '14 at 21:11

• I only repeated your construction to note that one can actually do it with fewer groups: take only the symmetric groups, or the $GL$s or the altenrating groups of... – Mariano Suárez-Álvarez Jan 7 '14 at 21:06