### 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 language | English |
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Pages (from-to) | 165-179 |

Number of pages | 15 |

Journal | Discrete Mathematics |

Volume | 200 |

Issue number | 1-3 |

Publication status | Published - Apr 6 1999 |

### Fingerprint

### Keywords

- Density
- Indecomposable group
- Uniform factorization

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*200*(1-3), 165-179.

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

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 200, no. 1-3, pp. 165-179.

}

TY - JOUR

T1 - On the orders of directly indecomposable groups

AU - Erdős, P.

AU - Pálfy, Péter P.

PY - 1999/4/6

Y1 - 1999/4/6

N2 - 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.

AB - 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.

KW - Density

KW - Indecomposable group

KW - Uniform factorization

UR - http://www.scopus.com/inward/record.url?scp=0347670963&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347670963&partnerID=8YFLogxK

M3 - Article

VL - 200

SP - 165

EP - 179

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -