For a central subgroup $C$ of finite index of a group $G$ it is wellknown that the power map $$G \to C,\;g \mapsto g^{(G:C)}$$ is a group homomorphism. This is commonly proved by help of the transfer map (see for example the group theory book of Robinson (10.1.3) or Isaacs (5.6)). I wonder, however, if there is a direct proof that avoids using the transfer ?
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