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Let G be a finite group. A collection P = {H1; ... ;Hr} of subgroups of
G, where r > 1, is said a non-trivial partition of G if every non-identity element of
G belongs to one and only one Hi, for some 1 <=i<= r. We call a group G that does
not admit any non-trivial partition a partition-free group. In this paper, we study a
partition-free group G whose all proper non-cyclic subgroups admit non-trivial partitions.
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