We say that a finite group $G$ is conjugacy expansive if for any normal subset $S$ and any conjugacy class $C$ of $G$ the normal set $SC$ consists of at least as many conjugacy classes of $G$ as $S$ does. Halasi, Mar'oti, Sidki, Bezerra have shown that a group is conjugacy expansive if and only if it is a direct product of conjugacy expansive simple or abelian groups. By considering a character analogue of the above, we say that a finite group $G$ is character expansive if for any complex character $\alpha$ and irreducible character $\chi$ of $G$ the character $\alpha \chi$ has at least as many irreducible constituents, counting without multiplicity, as $\alpha$ does. In this paper we take some initial steps in determining character expansive groups.

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