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Let $G$ be a finite abelian group and let $\exp(G)$ denote the least commonmultiple of the orders of all elements of $G$. A $BH(G,h)$ matrix is a$G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th rootsof unity such that $HH^*=|G|I_{|G|}$. In this paper, we study the relationbetween $G$ and $h$ so that a $BH(G,h)$ matrix exists. We will only focus on$BH(\mathbb{Z}_n,h)$ matrices and $BH(G,2p^b)$ matrices, where $p$ is an oddprime. By our results, there are $2687$ open cases left for the existence of$BH(\mathbb{Z}_n,h)$ matrices in which $1\leq n,h \leq 100$. In the lastsection, we show that $BH(\mathbb{Z}_n,h)$ matrices can be used to construct anew family of perfect polyphase arrays.

Necessary Conditions for the Existence of Group-Invariant Butson Matrices and a New Family of Perfect Arrays