On a result of S. Delsarte

Abstract:

For an isomorphism type of a finite abelian $p$-group $X$ it is shown that the matrix $({p^{\left \langle {s(D),s(Y)} \right \rangle }})$ is nonsingular; $D$, $Y \in \left \{ {S|S \leqslant X\;{\text {and}}\;S \ne X} \right \}$, the set of all proper isomorphism type of subgroups of $X$. Here $s(Y)$ denotes the signature of $Y$. This completes the proof of a result of ${\text {S}}$. Delsarte which gives explicit formulas for the number of automorphisms of $X$, the number of subgroups of $X$ isomorphic to $Y$ (and the number of homomorphisms from $Y$ into $X$) in terms of signatures.