Alternating groups as products of four conjugacy classes
- PDF / 345,504 Bytes
- 10 Pages / 439.37 x 666.142 pts Page_size
- 52 Downloads / 206 Views
Archiv der Mathematik
Alternating groups as products of four conjugacy classes Martino Garonzi
´ ti and Attila Maro
Abstract. Let G be the alternating group Alt(n) on n letters. We prove that for any ε > 0, there exists N = N (ε) ∈ N such that whenever n ≥ N and A, B, C, D are normal subsets of G each of size at least |G|1/2+ε , then ABCD = G. Mathematics Subject Classification. Primary 20E45; Secondary 20B30. Keywords. Alternating group, Conjugacy class, Character sum.
1. Introduction. Given two subsets A, B of a group G, we denote by AB the set of products ab where a ∈ A, b ∈ B. A subset A of G is called normal if gAg −1 = A for all g ∈ G. Clearly, a subset of G is normal if and only if it is a union of conjugacy classes. Observe that if A and B are normal sets, then AB = BA. The covering number of a nontrivial conjugacy class C of a finite nonabelian simple group G is the minimum positive integer k such that C k = G. Brenner [2] showed that almost all conjugacy classes of the alternating group Alt(n) have covering number at most 4, and observed that there are classes with covering number 4, for example, the class of fixed-point-free involutions (see the penultimate paragraph of the introduction). Larsen and Shalev [5, Theorem 1.13] proved that an element g of the symmetric group Sym(n) satisfies (g Sym(n) )2 = Alt(n) with probability tending to 1 as n → ∞. (Here and throughout the paper xG denotes the conjugacy class of an element x in a finite group G.) For a related result, see [5, Theorem 1.20]. Larsen and Shalev also proved [5, Theorem 1.14] that if n is sufficiently large, then any element g ∈ Sym(n) with at most n/5 fixed points satisfies (g Sym(n) )4 = Alt(n). In this paper, we take a different approach, considering the product of possibly distinct normal sets. Larsen, Shalev, and Tiep [6] proved that if ε > 0 is a constant, then for sufficiently large n, the following holds: whenever A, B
´ ti M. Garonzi and A. Maro
Arch. Math.
are two normal subsets of G = Alt(n) of size larger than ε|G|, then AB contains every nontrivial element of G, and they proved that the same holds for simple groups of Lie type of bounded rank. In this context, a subset is large if it has size at least the size of G multiplied by a universal positive constant (less than 1). Observe that, using their result, it is easy to see that, if A, B, C are large normal subsets of G, then ABC = G. We are interested in studying largeness related to the size of G raised to a constant γ. Let G = Alt(n) be the alternating group on n letters. In [7, Theorem 1.3], it is proved that there exists γ with 0 < γ < 1 such that whenever 8 normal subsets of G have size at least |G|γ , their product is G. It was asked if the same holds with less than 8 normal sets. In this paper, we prove that the result holds for 4 normal sets and that if γ is close to 1/2, then this is best possible. Theorem 1.1. For any ε > 0, there exists N ∈ N such that whenever n > N and A, B, C, D are normal subsets of G = Alt(n) such that all of the numbers |A||B|, |A||C|, |A||D|, |B||
Data Loading...
