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' 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.