On computing the number of subgroups of a finite Abelian group

Abstract

We consider the numberN A (r) of subgroups of orderp r ofA, whereA is a finite Abelianp-group of type α=α12,...,α l (α)), i.e. the direct sum of cyclic groups of order ααii. Formulas for computingN A (r) are well known. Here we derive a recurrence relation forN A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN A (r) and the Gaussian binomial coefficient\(\left[ {\begin{array}{*{20}c} {l(\alpha ) + r - 1} \\ r \\ \end{array} } \right]\).

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Stehling, T. On computing the number of subgroups of a finite Abelian group. Combinatorica 12, 475–479 (1992). https://doi.org/10.1007/BF01305239

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AMS subject classification code (1991)

  • 05 A 15
  • 20 K 01