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On a result of S. Delsarte

Authors: Gregory Constantine and Ravi S. Kulkarni
Journal: Proc. Amer. Math. Soc. 92 (1984), 149-152
MSC: Primary 20B25; Secondary 05A15, 20K30
MathSciNet review: 749907
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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\vert 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 [Enhancements On Off] (What's this?)

  • [1] S. Delsarte, Fonctions de Möbius sur les groupes abeliens finis, Ann. of Math. (2) 49 (1948), 600–609 (French). MR 0025463,
  • [2] I. Schur, Bemerkungen zur Theorie der beschränkten Bilinearformen suit unendlich vielen Veränderlichen, J. Reine Angew. Math. 140 (1911), 1-28.
  • [3] G. Constantine, Topics in combinatorics, Unpublished Manuscript, Indiana University, 1982.

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Keywords: Signature of a finite abelian $ p$-group, Möbius inversion, generating function
Article copyright: © Copyright 1984 American Mathematical Society

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