Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Countable-configurations and paradoxical decompositions   In this paper we define countable-configuration of groups and prove that two
Hopfian groups with the same set of countable-configurations are isomorphic and
vice versa. We also study the countable paradoxical decomposition of groups. It
is proved that a group G admits a countable paradoxical decomposition if and
only if it is infinite.

28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph