## Abstract

The Kurosh rank r_{K} (H) of a subgroup H of a free product ∏_{α ∈ I}^{*} G_{α} of groups G_{α}, α ∈ I, is defined accordingly to the classic Kurosh subgroup theorem as the number of free factors of H. We prove that if H_{1}, H_{2} are subgroups of ∏_{α ∈ I}^{*} G_{α} and H_{1}, H_{2} have finite Kurosh rank, then over(r, ̄)_{K} (H_{1} ∩ H_{2}) ≤ 2 frac(q_{*}, q^{*} - 2) over(r, ̄)_{K} (H_{1}) over(r, ̄)_{K} (H_{2}) ≤ 6 over(r, ̄)_{K} (H_{1}) over(r, ̄)_{K} (H_{2}), where over(r, ̄)_{K} (H) = max (r_{K} (H) - 1, 0), q^{*} is the minimum of orders >2 of finite subgroups of groups G_{α}, α ∈ I, q^{*} : = ∞ if there are no such subgroups, and frac(q^{*}, q^{*} - 2) : = 1 if q^{*} = ∞. In particular, if the factors G_{α}, α ∈ I, are torsion-free groups, then over(r, ̄)_{K} (H_{1} ∩ H_{2}) ≤ 2 over(r, ̄)_{K} (H_{1}) over(r, ̄)_{K} (H_{2}).

Original language | English (US) |
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Pages (from-to) | 465-484 |

Number of pages | 20 |

Journal | Advances in Mathematics |

Volume | 218 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2008 |

## Keywords

- Free products of groups
- Kurosh rank of a subgroup of a free product
- Subgroups of free products

## ASJC Scopus subject areas

- General Mathematics