On the convergence and periods in Boolean group algebras (Q1964723)

Let \(B\) be the Boolean ring \(\{ 0,1\}\), and consider the group algebra \(BG\) where \(G\) is an abelian group of order \(n\). The support \(\text{supp }a\) of an element \(a\in BG\) is defined to be the set of elements in \(G\) whose coefficients in \(a\) are nonzero. The paper proves a number of results which relate \(\text{supp }a\) to the multiplicative properties of \(a\). For example, some power of \(a\) is equal to \(\sum_{g\in G}g\) if and only if \(\langle g^{-1}\text{supp }a\rangle =G\) for some \(g\in \text{supp }a\) (Theorem 2.1); and (for \(a\neq 0\)) there exists some integer \(m\geq 0\) such that \(a^{m}=a^{m+1}\) if and only if \(g\in \langle g^{-1}\text{supp }a\rangle \) for each \(g\in \text{supp }a\) (Theorem 3.5). Some applications to circulant Boolean matrices are given.