Product decompositions of quasirandom groups and a Jordan type theorem
Product decompositions of quasirandom groups and a Jordan type theorem
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拟随机群的乘积分解和乔丹型定理
DOI：
10.4171/jems/275
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发表时间：
2007
影响因子：
2.6
通讯作者：
L. Pyber
中科院分区：
文献类型：
--
作者：
N. Nikolov;L. Pyber
我们首先注意到Gowers关于群中无积集的一个结果有一个意想不到的结果：如果k是有限群G的表示的最小度，则对于G的每个子集B，|B| > |G|我们有B^3 = G。
我们用它来获得改进版本的最近深定理的Helfgott和Shalev关于产品分解的有限简单的群体，更简单的证明。
另一方面，我们证明了Jordan定理的一个版本，它意味着如果k>1，则对于某个绝对常数c，G有一个指数至多为ck ^2的真子群，因此G有一个大小至少为$的无积子集。|G|/ c'k$。这回答了Gowers的一个问题。
We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 = G.
We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs.
On the other hand, we prove a version of Jordan's theorem which implies that if k>1, then G has a proper subgroup of index at most ck^2 for some absolute constant c, hence a product-free subset of size at least $|G| / c'k$. This answers a question of Gowers.