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We say that a finite group $G$ is conjugacy expansive if for anynormal subset $S$ and any conjugacy class $C$ of $G$ the normalset $SC$ consists of at least as many conjugacy classes of $G$ as$S$ does. Halasi, Mar'oti, Sidki, Bezerra have shown that a groupis conjugacy expansive if and only if it is a direct product ofconjugacy expansive simple or abelian groups.By considering a character analogue of the above, we say that afinite group $G$ is character expansive if for any complexcharacter $alpha$ and irreducible character $chi$ of $G$ thecharacter $alpha chi$ has at least as many irreducibleconstituents, counting without multiplicity, as $alpha$ does. Inthis paper we take some initial steps in determining characterexpansive groups.