University of Isfahan International Journal of Group Theory 2251-7650 5 4 2016 12 01 A gap theorem for the ZL-amenability constant of a finite group 27 46 9562 10.22108/ijgt.2016.9562 EN Yemon Choi Lancaster University Journal Article 2014 10 19 It 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. https://ijgt.ui.ac.ir/article_9562_6ce2f0b168ba2560656bbdd6cd54eaae.pdf