On a result of S. Delsarte
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- by Gregory Constantine and Ravi S. Kulkarni PDF
- Proc. Amer. Math. Soc. 92 (1984), 149-152 Request permission
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.References
- S. Delsarte, Fonctions de Möbius sur les groupes abeliens finis, Ann. of Math. (2) 49 (1948), 600–609 (French). MR 25463, DOI 10.2307/1969047 I. Schur, Bemerkungen zur Theorie der beschränkten Bilinearformen suit unendlich vielen Veränderlichen, J. Reine Angew. Math. 140 (1911), 1-28. G. Constantine, Topics in combinatorics, Unpublished Manuscript, Indiana University, 1982.
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 149-152
- MSC: Primary 20B25; Secondary 05A15, 20K30
- DOI: https://doi.org/10.1090/S0002-9939-1984-0749907-7
- MathSciNet review: 749907