A group G has an abelian partition if it has a set theoretic partition into disjoint commutative subsets where for all i and the identity is in A(0), such a group is called an abelian partitionable group (group). The problem of classifying groups was recently taken up by Mahmoudifar et al. who classified all groups with n = 2, 3. The motivation for this problem can be found in graph theory where partitions of graphs into induced complete subgraphs is of great importance. We achieve a partial classification of groups and introduce a family of groups with no abelian partition. Communicated by Mark L. Lewis
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