A gap theorem for the ZL-amenability constant of a finite group

It was shown in [‎A‎. ‎Azimifard‎, ‎E‎. ‎Samei and N‎. ‎Spronk‎, ‎Amenability properties of the centres of group algebras‎, J‎. ‎Funct‎. ‎Anal.‎, ‎256 no‎. ‎5 (2009) 15441564‎.] that the ZLamenability 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 nonabelian 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.  ‎Here we show that if $G$ is finite and nonabelian then its ZLamenability constant is at least $7/4$‎, ‎which is known to be best possible‎. ‎We avoid use of Rider#039 s reslt‎, ‎by analyzing the cases where $G$ is just nonabelian‎, ‎using calculations from [‎M‎. ‎Alaghmandan‎, ‎Y‎. ‎Choi and E‎. ‎Samei‎, ‎ZLamenability constants of finite groups with two character degrees‎, Canad‎. ‎Math‎. ‎Bull.‎, ‎57 (2014) 449462‎.]‎, ‎and establishing a new estimate for groups with trivial centre‎.