A note on Oliver's p-group conjecture

Let S be a finite p-group for an odd prime p, Oliver proposed the conjecture that the Thompson subgroup J(S) is always contained in the Oliver subgroup X(S). That means he conjectured that |J(S)X(S):X(S)|=1. Let X1(S) be a subgroup of S such that X1(S)/X(S) is the center of S/X(S). In this short note, we prove that J(S)≤X(S) if and only if J(S)≤X1(S). As an easy application, we prove that |J(S)X(S):X(S)|≠p.