We study Killing forms on finite groups arising from the extension of the theory of Killing forms on Lie algebras to braided-Lie algebras. For braided structures associated with finite groups, these forms admit an expression in terms of the character of the conjugation action on a certain conjugation-stable subset of the group. Motivated by Cartan’s criterion and earlier work by López Peña, Majid, and Rietsch, we investigate the non-degeneracy of such Killing forms defined on conjugation-stable subsets of finite groups. We will particularly focus on the family of groups PSL(2,q) and on conjugacy classes of involutions in symmetric and alternating groups. Our methods reveal interesting connections with linear independency of permutation matrices, generation properties of vector spaces of complex-valued functions and fusion within conjugacy classes of finite groups. This talk is based on a paper in collaboration with Kevin Piterman and recent work with Carsten Dietzel.