A finite group G is called a Qi-group if all of its non-linear irreducible characters are rational valued and G is called a Q-group if all of its irreducible characters are rational valued. Obviously every Q-group is a Qi-group. A finite group which is a Q_1-group but not a Q-group is called a Q_1'-group. In this paper some properties of non-abelian Q_1'-groups are investigated. In particular, under certain conditions we find a relationship between a Q_1'-group and a group with a Camina pair.
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