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On the number of permutations arising from a problem in cell-biology.

Dorninger, D., Hule, H. (1991)

Mathematica Pannonica

On the number of rim hook tableaux

С.В. Фомин, Н. Лулов (1995)

Zapiski naucnych seminarov POMI

On the number of solutions of equation $x^{p^{k}} = 1$ in a finite group

Yakov Berkovich (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Theorem A yields the condition under which the number of solutions of equation $x^{p^{k}} = 1$ in a finite $p$ -group is divisible by $p^{n + k}$ (here $n$ is a fixed positive integer). Theorem B which is due to Avinoam Mann generalizes the counting part of the Sylow Theorem. We show in Theorems C and D that congruences for the number of cyclic subgroups of order $p^{k}$ which are true for abelian groups hold for more general finite groups (for example for groups with abelian Sylow $p$ -subgroups).

On the number of subgroups of finite abelian groups

Aleksandar Ivić (1997)

Journal de théorie des nombres de Bordeaux

Let $T (x) = K_{1} x {log}^{2} x + K_{2} x log x + K_{3} x + Δ (x),$ where $T (x)$ denotes the number of subgroups of all abelian groups whose order does not exceed $x$ and whose rank does not exceed $2$ , and $Δ (x)$ is the error term. It is proved that $\int_{1}^{X} Δ^{2} (x) d x ≪ X^{2} {log}^{31 / 3} X, \int_{1}^{X} Δ^{2} (x) d x = Ω (X^{2} {log}^{4} X) .$

On the number of subgroups of given index in the modular group.

W. Imrich, R. Razen (1979)

Monatshefte für Mathematik

On the Occurrence of the Complete Graph $K_{5}$ in the Hasse Graph of a Finite Group

Roland Schmidt (2006)

Rendiconti del Seminario Matematico della Università di Padova

On the Olson and the Strong Davenport constants

Oscar Ordaz, Andreas Philipp, Irene Santos, Wolfgang A. Schmid (2011)

Journal de Théorie des Nombres de Bordeaux

A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$ -groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general...