### Abstract

If V is a (possibly infinite) set, G a permutation group on V, ν ∈ V, and Ω is an orbit of the stabiliser G _{ν}, let G ^{Ω} _{ν} denote the permutation group induced by the action of G _{ν} on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G _{ν} and G ^{Ω} _{ν}. If G is primitive and G _{ν} is finite, then by a theorem of Betten et al. (J Group Theory 6:415-420, 2003) we can conclude that every composition factor of the group G _{ν} is also a composition factor of the group G ^{Ω(ν)} _{ν}. In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions that can be expressed in terms of the natural permutation topology on Sym(V). In particular, we show the following: If Ω = u ^{Gν} is a suborbit of a transitive closed subgroup G of Sym(V) with a normalizing overgroup N ≤ N _{Sym(V)}(G) such that the N-orbital {(ν ^{g}, u ^{g}) {pipe} u ∈ Ω, g ∈ N} is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G _{ν} is also a section of G ^{Ω} _{ν}. To demonstrate that the topological assumptions on G and the simple sections of G _{ν} cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G _{ν} is isomorphic to the modular group PSL(2, ℤ)≅ C _{2}*C _{3}, which is known to have infinitely many finite simple groups among its sections.

Original language | English (US) |
---|---|

Pages (from-to) | 497-504 |

Number of pages | 8 |

Journal | Monatshefte fur Mathematik |

Volume | 166 |

Issue number | 3-4 |

DOIs | |

State | Published - Jun 2012 |

### Fingerprint

### Keywords

- Locally-compact group
- Permutation group
- Point-stabiliser

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Monatshefte fur Mathematik*,

*166*(3-4), 497-504. https://doi.org/10.1007/s00605-010-0282-0

**On the point-stabiliser in a transitive permutation group.** / Potočnik, P.; Wilson, Stephen E.

Research output: Contribution to journal › Article

*Monatshefte fur Mathematik*, vol. 166, no. 3-4, pp. 497-504. https://doi.org/10.1007/s00605-010-0282-0

}

TY - JOUR

T1 - On the point-stabiliser in a transitive permutation group

AU - Potočnik, P.

AU - Wilson, Stephen E

PY - 2012/6

Y1 - 2012/6

N2 - If V is a (possibly infinite) set, G a permutation group on V, ν ∈ V, and Ω is an orbit of the stabiliser G ν, let G Ω ν denote the permutation group induced by the action of G ν on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G ν and G Ω ν. If G is primitive and G ν is finite, then by a theorem of Betten et al. (J Group Theory 6:415-420, 2003) we can conclude that every composition factor of the group G ν is also a composition factor of the group G Ω(ν) ν. In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions that can be expressed in terms of the natural permutation topology on Sym(V). In particular, we show the following: If Ω = u Gν is a suborbit of a transitive closed subgroup G of Sym(V) with a normalizing overgroup N ≤ N Sym(V)(G) such that the N-orbital {(ν g, u g) {pipe} u ∈ Ω, g ∈ N} is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G ν is also a section of G Ω ν. To demonstrate that the topological assumptions on G and the simple sections of G ν cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G ν is isomorphic to the modular group PSL(2, ℤ)≅ C 2*C 3, which is known to have infinitely many finite simple groups among its sections.

AB - If V is a (possibly infinite) set, G a permutation group on V, ν ∈ V, and Ω is an orbit of the stabiliser G ν, let G Ω ν denote the permutation group induced by the action of G ν on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G ν and G Ω ν. If G is primitive and G ν is finite, then by a theorem of Betten et al. (J Group Theory 6:415-420, 2003) we can conclude that every composition factor of the group G ν is also a composition factor of the group G Ω(ν) ν. In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions that can be expressed in terms of the natural permutation topology on Sym(V). In particular, we show the following: If Ω = u Gν is a suborbit of a transitive closed subgroup G of Sym(V) with a normalizing overgroup N ≤ N Sym(V)(G) such that the N-orbital {(ν g, u g) {pipe} u ∈ Ω, g ∈ N} is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G ν is also a section of G Ω ν. To demonstrate that the topological assumptions on G and the simple sections of G ν cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G ν is isomorphic to the modular group PSL(2, ℤ)≅ C 2*C 3, which is known to have infinitely many finite simple groups among its sections.

KW - Locally-compact group

KW - Permutation group

KW - Point-stabiliser

UR - http://www.scopus.com/inward/record.url?scp=84861648477&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861648477&partnerID=8YFLogxK

U2 - 10.1007/s00605-010-0282-0

DO - 10.1007/s00605-010-0282-0

M3 - Article

AN - SCOPUS:84861648477

VL - 166

SP - 497

EP - 504

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 3-4

ER -