The authors address the following question: If a finite group \(A\) acts on a finite \(p\)-group \(P\), what conditions guarantee that \(A\) contains a nontrivial normal subgroup \(H\) (possibly \(H=A\)) that acts trivially on \(P\)?
Let \(\mathfrak A_{p^2-1}\) be the formation of all finite abelian groups with exponent dividing \(p^2-1\). The authors elaborate on a result by Y. Berkovich and I. M. Isaacs in [J. Algebra 414, 82–94 (2014; Zbl 1316.20012)] and show the following. Let \(p\) be a prime and \(e\geq4\) (or \(e\geq5\) if \(p=2\)) an integer. Suppose that an odd-order group \(A\) acts on a finite \(p\)-group \(P\) of order greater than \(p^e\) such that every non-metacyclic subgroup of order \(p^e\) is stabilized by \(O^p(A)\). Then, \(P\) is centralized by \(O^p(A^{\mathfrak A_{p^2-1}})\), where \(A^{\mathfrak A_{p^2-1}}\) is the smallest normal subgroup of \(A\) such that \(A/A^{\mathfrak A_{p^2-1}}\in\mathfrak A_{p^2-1}\). They also prove that under the same assumptions, if \(\mathfrak Z_p\) is a set containing exactly one Sylow \(q\)-subgroup of \(A\) for each prime \(q\neq p\) dividing \(|A|\) and if every non-metacyclic subgroup of \(P\) of order \(p^e\) is stabilized by every element of \(\mathfrak Z_p\), then \(P\) is centralized by \(O^p(A^{\mathfrak A_{p^2-1}})\).
The \(p\)-length \(l_p(G)\) of a \(p\)-solvable group \(G\) is the length of the upper \(p'p\)-series of \(G\), and the \(p\)-rank \(r_p(G)\) of \(G\) is the maximal rank of all the \(p\)-chief factors of \(G\). As an application of the second result above, the authors prove that if \(G\) is a group of odd order with Sylow \(p\)-subgroup \(P\) of order greater than \(p^e\) for a fixed integer \(e\geq4\) and if every non-metacyclic subgroup of \(P\) of order \(p^e\) is permutable with every element of \(\mathfrak Z_p\) in \(G\), then \(l_p(G)\leq1\) and \(r_p(G)\geq2\). Moreover, if \(p\) is a Mersenne prime, then \(G\) is \(p\)-supersolvable. The last main result in the article asserts the following. Let \(G\) be a group of odd order and fix an integer \(e\geq4\). Let \(H\) be a normal subgroup of \(G\) and assume that \(P\) is a Sylow \(p\)-subgroup of \(H\) with \(|P|>p^e\). If every non-metacyclic subgroup of \(P\) of order \(p^e\) is permutable with every element of \(\mathfrak Z_p\) in \(G\), then \(H\) is contained the \(\mathfrak I^2_p(G)\)-hypercenter of \(G\), where \(\mathfrak I^2_p(G)\) is the formation of all groups of odd order and \(p\)-rank at most \(2\).