### Abstract

A subset {x
_{1},x
_{2},…,x
_{d}} of a group G invariably generates G if {x
_{1}
^{g
1
},x
_{2}
^{g
2
},…,x
_{d}
^{g
d
}} generates G for every d-tuple (g
_{1},g
_{2}…,g
_{d})∈G
^{d}. We prove that a finite completely reducible linear group of dimension n can be invariably generated by ⌊[Formula presented]⌋ elements. We also prove tighter bounds when the field in question has order 2 or 3. Finally, we prove that a transitive [respectively primitive] permutation group of degree n≥2 [resp. n≥3] can be invariably generated by O([Formula presented]) [resp. O([Formula presented])] elements.

Original language | English |
---|---|

Pages (from-to) | 250-289 |

Number of pages | 40 |

Journal | Journal of Algebra |

Volume | 524 |

Early online date | 4 Feb 2019 |

DOIs | |

Publication status | Published - 15 Apr 2019 |

### Keywords

- Conjugation properties in finite groups
- Generation of finite groups

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Invariable generation of permutation and linear groups.** / Tracey, Gareth.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 524, pp. 250-289. https://doi.org/10.1016/j.jalgebra.2019.01.018

}

TY - JOUR

T1 - Invariable generation of permutation and linear groups

AU - Tracey, Gareth

PY - 2019/4/15

Y1 - 2019/4/15

N2 - A subset {x 1,x 2,…,x d} of a group G invariably generates G if {x 1 g 1 ,x 2 g 2 ,…,x d g d } generates G for every d-tuple (g 1,g 2…,g d)∈G d. We prove that a finite completely reducible linear group of dimension n can be invariably generated by ⌊[Formula presented]⌋ elements. We also prove tighter bounds when the field in question has order 2 or 3. Finally, we prove that a transitive [respectively primitive] permutation group of degree n≥2 [resp. n≥3] can be invariably generated by O([Formula presented]) [resp. O([Formula presented])] elements.

AB - A subset {x 1,x 2,…,x d} of a group G invariably generates G if {x 1 g 1 ,x 2 g 2 ,…,x d g d } generates G for every d-tuple (g 1,g 2…,g d)∈G d. We prove that a finite completely reducible linear group of dimension n can be invariably generated by ⌊[Formula presented]⌋ elements. We also prove tighter bounds when the field in question has order 2 or 3. Finally, we prove that a transitive [respectively primitive] permutation group of degree n≥2 [resp. n≥3] can be invariably generated by O([Formula presented]) [resp. O([Formula presented])] elements.

KW - Conjugation properties in finite groups

KW - Generation of finite groups

UR - http://www.scopus.com/inward/record.url?scp=85061059911&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2019.01.018

DO - 10.1016/j.jalgebra.2019.01.018

M3 - Article

VL - 524

SP - 250

EP - 289

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -