## Abstract

We introduce the notion of the depth of a finite group G, defined as the minimal length of an unrefinable chain of subgroups from G to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups. We determine the simple groups of minimal depth, and show, somewhat surprisingly, that alternating groups have bounded depth. We also establish general upper bounds on the depth of simple groups of Lie type, and study the relation between the depth and the much studied notion of the length of simple groups. The proofs of our main theorems depend (among other tools) on a deep number-theoretic result, namely, Helfgott’s recent solution of the ternary Goldbach conjecture.

Original language | American English |
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Pages (from-to) | 2343-2358 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 6 |

DOIs | |

State | Published - 2018 |

### Bibliographical note

Funding Information:Received by the editors August 2, 2017, and, in revised form, August 21, 2017. 2010 Mathematics Subject Classification. Primary 20E32, 20E15; Secondary 20E28. The first and third authors acknowledge the hospitality and support of Imperial College, London, while part of this work was carried out. The third author acknowledges the support of ISF grant 1117/13 and the Vinik chair of mathematics which he holds.

Publisher Copyright:

© 2018 Amerian Mathematial Soiety.