Abstract
We consider the numberN A (r) of subgroups of orderp r ofA, whereA is a finite Abelianp-group of type α=α1,α2,...,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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