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The classification theorem for surfaces says that the complete set of homeomorphism classes of surfaces is { $S_g : g \geq 0$ } $ \cup$ { $N_k : k \geq 1$ }, where $S_g$ is a sphere with $g$ handle …

To expand on Steve D's comment, the answer is indeed yes for finite abelian groups. The following is a simplified version of an earlier proof (rendering some of the below comments obsolete). Proof. …

According to the wikipedia page, every group is indeed the automorphism group of some graph. This was proven independently in de Groot, J. (1959), Groups represented by homeomorphism groups, Mathem …

One way in which $\sum_A$ can be small is if $A$ is a small subgroup of $G$. To exclude such examples, define $X$ to be aperiodic, if the only solution to $X+x=X$ is $x=0$. DeVos, Goddyn, Mohar and …

Here's a nice proof of the Cantor-Bernstein theorem in the language of infinite graphs. Theorem. Let $G$ be an infinite graph with bipartition $(A,B)$. If $G$ has a matching saturating $A$ and a m …

For (1), the answer for finite permutations (as defined by the OP) is clearly yes. This of course is a characterization of finite permutations. A permutation is finite if and only if it is the produ …