p-Group Camina pairs

Abstract

Let $(G,Z(G))$ be a Camina pair. We prove that $G$ must be a $p$-group for some prime $p$. We also prove that $|Z(G)|<|G:Z(G){|}^{3/4}$. Also, we discuss how one might build examples with $|Z(G)|>|G:Z(G){|}^{1/2}$, although we are not able to prove the existence of such examples.

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