## Abstract

There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of n-fold coverings over the orientable Euclidean manifolds G_{3} and G_{5}, and calculate the numbers of non-equivalent coverings of each type. The manifolds G_{3} and G_{5} are uniquely determined among other forms by their homology groups H_{1}(G_{3})=Z_{3}×Z and H_{1}(G_{5})=Z. We classify subgroups in the fundamental groups π_{1}(G_{3}) and π_{1}(G_{5}) up to isomorphism. Given index n, we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating functions for the above sequences.

Original language | English |
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Pages (from-to) | 48-66 |

Number of pages | 19 |

Journal | Journal of Algebra |

Volume | 560 |

DOIs | |

Publication status | Published - 15 Oct 2020 |

## Keywords

- Crystallographic group
- Euclidean form
- Flat 3-manifold
- Non-equivalent coverings
- Platycosm
- NONEQUIVALENT COVERINGS
- REPRESENTATIONS
- SUBGROUPS
- SURFACES