A note on Oliver's \(p\)-group conjecture (Q2636374)

For a finite \(p\)-group \(S\) with an odd prime \(p\), \textit{B. Oliver} [Math. Proc. Camb. Philos. Soc. 137, No. 2, 321--347 (2004; Zbl 1077.55006)] conjectured that the Thompson subgroup \(J(S)\) is always contained in the Oliver subgroup \(X(S)\). That means that \(|J(S)X(S):X(S)|=1\). Let \(X_1(S)\) be a subgroup of \(S\) such that \(X_1(S)/X(S)\) is the center of \(S/X(S)\). The author proved that \(J(S)\leq X(S)\) if and only if \(J(S)\leq X_1(S)\).