A characterization of cyclic groups
If \((H)\) denotes the conjugacy class of a subgroup H of a finite group G, let \(a_{(H),(K)}=| H\setminus G/K|\), the number of double cosets. Let \(A\) be the square matrix \((a_{(H),(K)})\), indexed by the set of conjugacy classes of subgroups of \(G\). It is shown that \(\det(A)\neq 0\) if and only if \(G\) is cyclic. In fact the rank of \(A\) is the number of conjugacy classes of cyclic subgroups of \(G\). Moreover if \(G\) is cyclic of order \(n\), then \(\det (A)=\prod _{d\mid n}\varphi(d)\), where \(\varphi\) denotes the Euler function.
numbers of conjugacy classes of cyclic subgroups
scientific article; zbMATH DE number 4036061