Large Product-Free Subsets of Finite Groups

✍ Scribed by Kiran S. Kedlaya

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 476 KB
• Volume: 77
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2715

✦ Synopsis

A subset of a group is said to be product-free if the product of two of its elements is never itself an element of the subset. Using the classification of finite simple groups, we prove that every finite group of order n has a product-free subset of more than cn 11Â14 elements, for some fixed c>0. This improves on a lower bound of Babai and So s.

1997 Academic Press

Let G be a finite group of order n. A subset S of G is said to be product-free if, for any x, y # S (not necessarily distinct), xy  S. Define :(G) to be the size of the largest product-free subset of G. In [1], Babai and So s gave a simple construction that, together with the classification of finite simple groups (CFSG), shows that :(G)>cn 4Â7 for some constant c>0. The purpose of the present paper is to improve this lower bound as follows.

Theorem. There exists a constant c>0 such that :(G )>cn 11Â14 .

We begin the proof of the theorem by recalling Lemma 7.5 of [1].

Proof. Let ,: G Ä GÂN be the canonical homomorphism. If S is product-free in GÂN, then , &1 (S) is product-free in G. K Thus to prove any bound of the form :(G )>cn t for t 1, it suffices to consider the finite simple groups. In the case G=ZÂnZ, the set [k, ..., 2k&1], where k=w(n+1)Â3x , shows that :(G )>cn. By Lemma 1, the linear lower bound also holds for all groups with cyclic factor groups, so in particular for all solvable groups.