Abstract
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) 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. 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, Canad. Math. Bull., 57 (2014) 449-462.], and establishing a new estimate for groups with trivial centre.
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