Product decompositions of quasirandom groups and a Jordan type theorem

By Nikolay Nikolov, László Pyber. 2007

108 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/e6ad02c7a0d35ab86827ca71aa4d6ffdeda33130

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Aner Shalev
2. Michael Larsen
3. P. P. Varj'u
4. E. Szab'o
5. László Pyber
6. Robert M. Guralnick
7. Timothy C. Burness

In Discrete Mathematics and Combinatorics

1. Aner Shalev
2. Michael Larsen
3. László Pyber
4. P. P. Varj'u
5. E. Szab'o
6. Robert M. Guralnick
7. Emmanuel Breuillard
8. Harald Andrés Helfgott

In Electrical and Electronic Engineering

1. Michael Larsen
2. P. P. Varj'u
3. Aner Shalev
4. E. Szab'o
5. László Pyber
6. Robert M. Guralnick
7. Andreas Thom

Citations in the corpus

• Growth in groups: ideas and perspectives (2013, unknown): “or (A∪A−1 ∪{e})k = G, where δ>0 and k≥ 1 are absolute constants. Here, as usual, “absolute constant” means “really a constant”; in particular, the constants δand kdo not depend on p. Nikolov-Pyber [NP11] (following Gowers [Gow08]; see also [BNP08]) showed one can replace (A∪A−1∪{e})k by A3. Kowal”
• Random walks in compact groups (2012, background): “We say that a profinite group G is (c, α)-quasirandom if for every irreducible unitary representation π of G, we have dimπ ≥ c[G : Ker(π)]α.”
• Growth in finite simple groups of Lie type of bounded rank (2010, result): “It was observed in [45] that a result of Gowers [25] implies that (b) holds for an arbitrary simple group of Lie type L with k = 3 for some δ(r) which depends only”
• Characters of symmetric groups: sharp bounds and applications (2008, methodology): “In [NP] Nikolov and Pyber use a method of Gowers [G] to prove results of the form W 3 = G where W is a large subset of a finite group G.”