## Abstract

The

In this paper we determine when the intransitive maximal subgroups of S

*generating graph*Γ (G) of a finite group G has vertex set the non-identity elements of G, with two elements adjacent exactly when they generate G. A*coclique*in a graph is an empty induced subgraph, so a coclique in Γ (G) is a subset of G such that no pair of elements generate G. A coclique is*maximal*if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of G form a coclique in Γ (G), but this coclique need not be maximal.In this paper we determine when the intransitive maximal subgroups of S

_{n}and A_{n}are maximal cocliques in the generating graph. In addition, we prove a conjecture of Cameron, Lucchini, and Roney-Dougal in the case of G = A_{n}and S_{n}, when n is prime and n ≠ q^{d}-1/q-1 for all prime powers q and d ≥ 2. Namely, we show that two elements of G have identical sets of neighbours in Γ (G) if and only if they belong to exactly the same maximal subgroups.Original language | English |
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Article number | 56879 |

Number of pages | 20 |

Journal | Combinatorial Theory |

Volume | 2 |

Issue number | 1 |

DOIs | |

Publication status | Published - 30 Mar 2022 |

## Keywords

- Generating graph
- Alternating groups
- Symmetric groups