AbstractApplying the method that we presented in [19], in this article we prove: “Let G be an elementary abelian p‐group. Let n = dn1. If d(≠ p) is a prime not dividing n1, and the order w of d mod p satisfies \documentclass{article}\pagestyle{empty}\begin{document}$ w > \frac{{d^2}}{3} $\end{document}, then the Second Multiplier Theorem holds without the assumption n1 > λ, except that only one case is yet undecided: w ≤ d2, and \documentclass{article}\pagestyle{empty}\begin{document}$ \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 < 2w), 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.