Finite groups all of whose small subgroups are pronormal (Q343270)

In Theorem 2.7, those finite \(p\)-groups \(G\) are classified such that whenever \(H<G\) satisfies \(| H| ^2p<| G| \) then \(H\) is normal in \(G\). Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is said to be \textit{pronormal} if the subgroup \(H\) and \(H^g\) are conjugate in \(\langle H,H^g\rangle\) for each \(g\in G\). Let a nonnilpotent group \(G\) have no nontrivial nilpotent direct factor and let \(p\) be a minimal prime divisor of \(| G| \). Suppose that \(G\) satisfies the following condition: Whenever \(H<G\) is such that \(| H| ^p<| G| \), then \(H\) is pronormal in \(G\). In Theorem 4.2 all such \(G\) are described.