A remark on association schemes on finite groups (Q1123260)

An admissible partition \(\{D_ 0,D_ 1,...,D_ n\}\) of a finite group G is one which satisfies (i) \(D_ 0=\{e\}\). (ii) For each i, there is a j such that \(D_ i^{-1}=D_ j\). (iii) For all i,j,k, \[ D_ iD_ j=\cup^{n}_{k=0}p^ k_{ij}D_ k. \] \textit{Siu Lun Ma} [PhD Thesis, Univ. Hongkong (1985)] proved that for such a partition of G, the relations \(R_ i=\{(g,gd_ i)\in G\times G|\) \(d_ i\in D_ i\), \(g\in G\}\) partition the set \(G\times G\) in such a way that \((G,\{R_ i\})\) is an association scheme on the set G. A proof is given. The authors provide a basic construction for admissible partitions. Irr(G) is the set of irreducible characters of G. Consider the group HC(G) of all permutations \(\pi\) of the set G for which \(\pi (e)=e\), \(\pi (g^{-1})=\pi (g)^{-1}\) and \(\chi\) \(\circ \pi \in Irr(G)\) for all \(\chi\in Irr(G)\), and let A be any subgroup of HC(G) containing all permutations induced by inner automorphisms of G. If \(D_ 0,D_ 1,...,D_ n\) with \(D_ 0=\{e\}\) are the orbits of A on G, it is shown that \(\{D_ 0,D_ 1,...,D_ n\}\) is an admissible partition of G. Attention is drawn to two special cases. First, if G is a normal subgroup of a group H, then the H-conjugacy classes of G form an admissible partition of G. Second, if m is an integer divisible by the order of every element of G and \(\Gamma\) is any subgroup of the group of units of \({\mathbb{Z}}/m\), then an admissible partition of G is formed by the equivalence classes of the equivalence relation on G defined by \(x=g^{- 1}y^ tg\) for some \(g\in G\) and some \(t\in \Gamma\).