Let G be a non-trivial group. There are many possible ways for associating a graph with G, for the purpose of investigating this group using properties of the associated graph. In this paper we investigate the following graph GammaM(G), which is associated with the maximal subgroups of G. The vertices of GammaM(G) are the maximal subgroups of G, and we join two distinct vertices M_1, M_2, whenever the intersection of M_1 and M_2 is non-trivial. This graph will be called the maximal graph of G . First, we show that if G is a finite simple group, then GammaM(G) is connected and the diameter of GammaM(G), denoted by diam(GammaM(G)), is bounded. Our bound, 62, for diam(GammaM(G)) in the finite simple case is most certainly not best possible. We also show, that if G is a finite group, then GammaM(G) is either connected, or it has at least two vertices and no edges. Finite groups G with a non-connected graph GammaM(G) are classified. They are all solvable groups and if G is a finite solvable group with a connected graph GammaM(G), then the diameter of GammaM(G) is at most 2. There exist finitely generated infinite simple groups G with non-connected graphs GammaM(G). Such are, for example, the Tarski monsters, in which all maximal subgroups are of prime order. We determine the structure of finitely generated infinite non-simple groups G with a non-connected graph GammaM(G). In particular we show that if G is a finitely generated locally graded group with a non-connected graph GammaM(G), then G must be finite. In this paper we investigate the following graph GammaM(G), which is associated with the maximal subgroups of G. The vertices of GammaM(G) are the maximal subgroups of G, and we join two distinct vertices M_1, M_2, whenever the intersection of M_1 and M_2 is non-trivial. This graph will be called the maximal graph of G . First, we show that if G is a finite simple group, then GammaM(G) is connected and the diameter of GammaM(G), denoted by diam(GammaM(G)), is bounded. Our bound, 62, for diam(GammaM(G)) in the finite simple case is most certainly not best possible. We also show, that if G is a finite group, then GammaM(G) is either connected, or it has at least two vertices and no edges. Finite groups G with a non-connected graph GammaM(G) are classified. They are all solvable groups and if G is a finite solvable group with a connected graph GammaM(G), then the diameter of GammaM(G) is at most 2. There exist finitely generated infinite simple groups G with non-connected graphs GammaM(G). Such are, for example, the Tarski monsters, in which all maximal subgroups are of prime order. We determine the structure of finitely generated infinite non-simple groups G with a non-connected graph GammaM(G). In particular we show that if G is a finitely generated locally graded group with a non-connected graph GammaM(G), then G must be finite.

`http://hdl.handle.net/11386/3015834`

Titolo: | On a graph related to the maximal subgroups of a group |

Autori interni: | LONGOBARDI, Patrizia MAJ, Mercede |

Data di pubblicazione: | 2010 |

Rivista: | BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY |

Abstract: | Let G be a non-trivial group. There are many possible ways for associating a graph with G, for the purpose of investigating this group using properties of the associated graph. In this paper we investigate the following graph GammaM(G), which is associated with the maximal subgroups of G. The vertices of GammaM(G) are the maximal subgroups of G, and we join two distinct vertices M_1, M_2, whenever the intersection of M_1 and M_2 is non-trivial. This graph will be called the maximal graph of G . First, we show that if G is a finite simple group, then GammaM(G) is connected and the diameter of GammaM(G), denoted by diam(GammaM(G)), is bounded. Our bound, 62, for diam(GammaM(G)) in the finite simple case is most certainly not best possible. We also show, that if G is a finite group, then GammaM(G) is either connected, or it has at least two vertices and no edges. Finite groups G with a non-connected graph GammaM(G) are classified. They are all solvable groups and if G is a finite solvable group with a connected graph GammaM(G), then the diameter of GammaM(G) is at most 2. There exist finitely generated infinite simple groups G with non-connected graphs GammaM(G). Such are, for example, the Tarski monsters, in which all maximal subgroups are of prime order. We determine the structure of finitely generated infinite non-simple groups G with a non-connected graph GammaM(G). In particular we show that if G is a finitely generated locally graded group with a non-connected graph GammaM(G), then G must be finite. In this paper we investigate the following graph GammaM(G), which is associated with the maximal subgroups of G. The vertices of GammaM(G) are the maximal subgroups of G, and we join two distinct vertices M_1, M_2, whenever the intersection of M_1 and M_2 is non-trivial. This graph will be called the maximal graph of G . First, we show that if G is a finite simple group, then GammaM(G) is connected and the diameter of GammaM(G), denoted by diam(GammaM(G)), is bounded. Our bound, 62, for diam(GammaM(G)) in the finite simple case is most certainly not best possible. We also show, that if G is a finite group, then GammaM(G) is either connected, or it has at least two vertices and no edges. Finite groups G with a non-connected graph GammaM(G) are classified. They are all solvable groups and if G is a finite solvable group with a connected graph GammaM(G), then the diameter of GammaM(G) is at most 2. There exist finitely generated infinite simple groups G with non-connected graphs GammaM(G). Such are, for example, the Tarski monsters, in which all maximal subgroups are of prime order. We determine the structure of finitely generated infinite non-simple groups G with a non-connected graph GammaM(G). In particular we show that if G is a finitely generated locally graded group with a non-connected graph GammaM(G), then G must be finite. |

Handle: | http://hdl.handle.net/11386/3015834 |

Appare nelle tipologie: | 1.1.2 Articolo su rivista con ISSN |