... union of orbits of StabSL(2,p) (v), so that |D| is either 2, p + 1, p + 2, 2p, or 2p + Furthermore, as the size of a block of a permutation group divides the degree of the permutation group, |D| = ... is odd, a Sylow 2-subgroup Q of SL(2, p) is a generalized quaternion group Furthermore, the unique element of Q of order 2, namely −I, is contained is every Sylow 2-subgroup of SL(2, p) and is ... ) (D2 ) is cyclic of order 2, soc(Aut(Γi )/D2 ) = PSL(2, p) is doubly-transitive, Γi is a Cayley digraph of the generalized quaternion group of order 2k+1 Proof In view of Theorem 2.2, we need...