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Let  G  be a finite group. In this paper we give a new necessary condition for two real  G -modules to be Smith equivalent if  G  has a normal Sylow 2-subgroup. We show that the condition is also sufficient under certain hypotheses. By results on the Smith equivalence obtained in this paper, the primary Smith sets are not subgroups of the real representation rings for various Oliver groups with normal Sylow 2-subgroups.

A necessary condition for the Smith equivalence of G -modules and its sufficiency