Japan. J. Math. 10, 43--96 (2015)

Appendix: On some Gelfand pairs and commutative association schemes

E. Bannai, H. Tanaka

Abstract: This paper is Appendix of the paper of T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli [8].

We pay close attention on a special condition related to Gelfand pairs. Namely, we call a finite group $G$ and its automorphism $\sigma$ satisfy Condition ($\bigstar$) if the following condition is satisfied: if for $x,y\in G$, $x\cdot x^{-\sigma}$ and $y\cdot y^{-\sigma}$ are conjugate in $G$, then they are conjugate in $K=C_G(\sigma)$. The main purpose of the note was to study the meanings of this condition, as well as showing many examples of $G$ and $\sigma$ which do (or do not) satisfy Condition ($\bigstar$).