## Abstract

Let G be a (finite or infinite) group such that G/Z(G) is not simple. The non-commuting, non-generating graph Ξ(G) of G has vertex set G\Z(G), with vertices x and y adjacent whenever [x, y] ̸= 1 and ⟨x, y⟩ ̸ = G. We investigate the relationship between the structure of G and the connectedness and diameter of Ξ(G). In particular, we prove that the graph either: (i) is connected with diameter at most 4; (ii) consists of isolated vertices and a connected component of diameter at most 4; or (iii) is the union of two connected components of diameter 2. We also describe in detail the finite groups with graphs of type (iii). In the companion paper [17], we consider the case where G/Z(G) is finite and simple.

Original language | English |
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Pages (from-to) | 1395-1418 |

Number of pages | 24 |

Journal | Algebraic Combinatorics |

Volume | 6 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2023 |

Externally published | Yes |

## Keywords

- generating graph
- graphs defined on groups
- non-commuting non-generating graph
- soluble groups