Submonoids of integer group determinants

For a finite group $G$, let $S(G)$ denote the set of all integer values obtained from the group determinant by substituting integers for its variables. Such values are called integer group determinants. Giving a complete description of $S(G)$ is a difficult problem, known as the Taussky-Todd integer group determinant problem. In this paper, we introduce a submonoid $S'(G)$ of the monoid $S(G)$ arising from a specific specialization of the group determinant, and provide a framework for investigating certain aspects of the lower structure of $S(G)$. For finite groups $G$ of order at most $5$, we determine $S'(G)$ explicitly and present concrete examples of integer group determinants.The motivation for considering the submonoid $S'(G)$ is that specialization patterns can retain meaningful arithmetic information. In particular, this viewpoint may be useful in contexts related to classical number-theoretic problems, such as the search for Wieferich primes.