## Abstract

Fix a positive integer d and let Γ_{d} be the class of finite groups without sections isomorphic to the alternating group A_{d}. The groups in Γ_{d} were studied by Babai, Cameron and Pálfy in the 1980s and they determined bounds on the order of a primitive permutation group with this property, which have found a wide range of applications. Subsequently, results on the base sizes of such groups were also obtained. In this paper we replace the structural conditions on the group by restrictions on its point stabilizers, and we obtain similar, and sometimes stronger conclusions. For example, we prove that there is a linear function f such that the base size of any finite primitive group with point stabilizers in Γ_{d} is at most f(d). This generalizes a recent result of the first author on primitive groups with solvable point stabilizers. For non-affine primitive groups we obtain stronger results, assuming only that stabilizers of c points lie in Γ_{d}. We also show that if G is any permutation group of degree n whose c-point stabilizers lie in Γ_{d}, then |G|⩽((1+o_{c}(1))d/e)^{n−1}. This asymptotically extends and improves a d^{n−1} upper bound on |G| obtained by Babai, Cameron and Pálfy assuming G∈Γ_{d}.

Original language | American English |
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Pages (from-to) | 160-185 |

Number of pages | 26 |

Journal | Journal of Algebra |

Volume | 607 |

DOIs | |

State | Published - 1 Oct 2022 |

### Bibliographical note

Funding Information:AS acknowledges the support of ISF grant 686/17 and the Vinik Chair of Mathematics which he holds. Both authors thank Joanna Fawcett and Michael Giudici for helpful comments on twisted wreath products. They also thank an anonymous referee for many insightful comments, corrections and suggestions on an earlier version of this paper.

Publisher Copyright:

© 2021 Elsevier Inc.

## Keywords

- Base sizes
- Primitive groups
- Restricted stabilizers