On the orders of directly indecomposable groups

P. Erdős, Péter P. Pálfy

Research output: Contribution to journalArticle

1 Citation (Scopus)

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
Publication statusPublished - Apr 6 1999

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Decomposition
Odd number
Connected Components
Graph in graph theory
Zero
Asymptotic Formula
Divisor
Exception
Decompose
Integer

Keywords

  • Density
  • Indecomposable group
  • Uniform factorization

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

On the orders of directly indecomposable groups. / Erdős, P.; Pálfy, Péter P.

In: Discrete Mathematics, Vol. 200, No. 1-3, 06.04.1999, p. 165-179.

Research output: Contribution to journalArticle

Erdős, P & Pálfy, PP 1999, 'On the orders of directly indecomposable groups', Discrete Mathematics, vol. 200, no. 1-3, pp. 165-179.
Erdős, P. ; Pálfy, Péter P. / On the orders of directly indecomposable groups. In: Discrete Mathematics. 1999 ; Vol. 200, No. 1-3. pp. 165-179.
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