On the orders of directly indecomposable groups

Paul Erdos, Péter P. Pálfy

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Abstract

We investigate the set of those integers n for which directly indecomposable groups of order n exist. For even n such groups are easily constructed. In contrast, we show that the density of the set of odd numbers with this property is zero. For each n we define a graph whose connected components describe uniform direct decompositions of all groups of order n. We prove that for almost all odd numbers (i.e., with the exception of a set of density zero) this graph has a single 'big' connected component and all other vertices are isolated. We also give an asymptotic formula for the number of isolated vertices of the graph, i.e., for the number of prime divisors q of n such that every group of order n has a cyclic direct factor of order q.

Original languageEnglish
Pages (from-to)165-179
Number of pages15
JournalDiscrete Mathematics
Volume200
Issue number1-3
DOIs
Publication statusPublished - Apr 6 1999

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Keywords

  • Density
  • Indecomposable group
  • Uniform factorization

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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