## Abstract

Let G be a nonabelian group and n a natural number. We say that G has a strict n-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup A and n nonempty subsets B-1, B-2, . . . , B-n, such that vertical bar B-i vertical bar > 1 for each i and within each set B-i, no two distinct elements commute. We show that every finite nonabelian group has a strict n-split decomposition for some n. We classify all finite groups G, up to isomorphism, which have a strict n-split decomposition for n = 1, 2, 3. Finally, we show that for a nonabelian group G having a strict n-split decomposition, the index vertical bar G : A vertical bar is bounded by some function of n.

Original language | English |
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Pages (from-to) | 1051-1082 |

Number of pages | 32 |

Journal | Taiwanese Journal of Mathematics |

Volume | 22 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Oct 2018 |

## Keywords

- strict n-split decomposition
- simple group
- commuting graph
- Commuting graph
- Simple group
- Strict n-split decomposition