The multiplier conjecture for elementary abelian groups

✍ Scribed by Qiu Weisheng

• Publisher: John Wiley and Sons
• Year: 1994
• Tongue: English
• Weight: 551 KB
• Volume: 2
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.3180020303

✦ Synopsis

Applying the method that we presented in , in this article we prove: "Let G be an elementary abelian p-group. Let n = dnl. If d(# p) is a prime not dividing nl, and the order w of d mod p satisfies w > 7 , then the Second Multiplier Theorem holds without the assumption nl > A, except that only one case is yet undecided: w 5 dZ, and 5 2 3, and t is a quadratic residue mod p, and t is not congruent to x 2~ ' (mod p) (1 I : j < 2w), where t is an integer meeting the conditions of Second Multiplier Theorem, and x is a primitive root dz P -1 e . O f p." 0 1994 John Wiley & Sons, Inc.

1 . INTRODUCTION

Definition 1. Let G be a group of order v, and D a k-subset of G, with 1 < k < v -1.

Then D is called a (v, k,A)-difference set if the list of "differences" x y -' ( x , y E D ) contains each nonidentity element of G exactly A times. If G is abelian (resp. cyclic), D is also called abelian (resp. cyclic). An easy counting argument shows that

Definition 2. Let D be a difference set in a finite group G. An automorphism r of G is called a multiplier for D if r sends D onto Dg for some g E G . If G is abelian and if r is *The project supported by National Natural Science Foundation of China.