University of IsfahanInternational Journal of Group Theory2251-76505420161201A gap theorem for the ZL-amenability constant of a finite group2746956210.22108/ijgt.2016.9562ENYemon ChoiLancaster UniversityJournal Article20141019It was shown in [A. Azimifard, E. Samei and N. Spronk, Amenability properties of the centres of group algebras, <em>J. Funct. Anal.</em>, <strong>256</strong> no. 5 (2009) 1544-1564.] that the ZL-amenability constant of a finite group is always at least $1$, with equality if and only if the group is abelian. It was also shown that for any finite non-abelian group this invariant is at least $301/300$, but the proof relies crucially on a deep result of D. A. Rider on norms of central idempotents in group algebras. <br /> Here we show that if $G$ is finite and non-abelian then its ZL-amenability constant is at least $7/4$, which is known to be best possible. We avoid use of Rider's reslt, by analyzing the cases where $G$ is just non-abelian, using calculations from [M. Alaghmandan, Y. Choi and E. Samei, ZL-amenability constants of finite groups with two character degrees, <em>Canad. Math. Bull.</em>, <strong>57</strong> (2014) 449-462.], and establishing a new estimate for groups with trivial centre.http://ijgt.ui.ac.ir/article_9562_6ce2f0b168ba2560656bbdd6cd54eaae.pdf