Existence of normal subgroups in finite \(p\)-groups. (Q2474484)

The following result is proved: Theorem. Suppose that \(S\) is a \(p\)-group, and \(B<S\) is of order \(p^n\) and \(n\leq p-1\). If \(\text{cl}(B)\leq c\) and \(p\geq n-2+(n/c)\), then there is in \(B^S\) an \(S\)-invariant subgroup \(B^*\) of order \(p^n\) and class \(\leq c\) such that \(\exp(B^*)\leq\exp(B)\). In particular, if \(c=1\) and \(p>2n-2\), then there is an \(S\)-invariant Abelian \(B^*\leq B^S\) of order \(p^n\) such that \(\exp(B^*)\leq\exp(B)\). This result is known for \(n\leq 5\) and \(p>2\).