Growth of products of subsets in finite simple groups

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-05-23 DOI:10.1112/blms.13093

Daniele Dona, Attila Maróti, László Pyber

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Abstract

We prove that the product of a subset and a normal subset inside any finite simple non-abelian group $G$ grows rapidly. More precisely, if $A$ and $B$ are two subsets with $B$ normal and neither of them is too large inside $G$ , then $| A B | ⩾ {| A | | B |}^{1 - ε}$ where $ε > 0$ can be taken arbitrarily small. This is a somewhat surprising strengthening of a theorem of Liebeck, Schul, and Shalev.

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有限简单群中子集积的增长

我们证明，在任何有限简单非阿贝尔群 G $G$ 内，子集与正常子集的乘积都会快速增长。更确切地说，如果 A $A$ 和 B $B$ 是两个子集，B $B$ 是正常子集，并且它们在 G $G$ 内都不是太大，那么 | A B | | | A | | B | 1 - ε $|AB| \geqslant |A||B|^{1-\epsilon }$ 其中 ε > 0 $\epsilon >0$ 可以任意取小。这是对 Liebeck、Schul 和 Shalev 的一个定理的加强，有点出人意料。

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