The multiplier conjecture for elementary abelian groups

Applying the method that we presented in [19], in this article we prove: “Let G be an elementary abelian p ‐group. Let n = dn 1 . If d(≠ p ) is a prime not dividing n 1 , and the order w of d mod p satisfies \documentclass{article}\pagestyle{empty}\begin{document}$ w > \frac{{d^2}}{3} $\end{document} , then the Second Multiplier Theorem holds without the assumption n 1 > λ, except that only one case is yet undecided: w ≤ d 2 , and \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{p - 1}}{{2w}} \ge 3 $\end{document} , and t is a quadratic residue mod p , and t is not congruent to \documentclass{article}\pagestyle{empty}\begin{document}$ x^{\frac{{p - 1}}{{2w}}j} $\end{document} (mod p ) (1 ≤ j < 2 w ), where t is an integer meeting the conditions of Second Multiplier Theorem, and x is a primitive root of p .”. © 1994 John Wiley & Sons, Inc.