An element of \(H^*(G,\mathbb{F}_p)\) is called essential if it restricts to zero on all proper subgroups \(H\) of \(G\), and \(\text{Ess}^*(G)\) denotes the ideal of \(H^*(G,\mathbb{F}_p)\) consisting of essential elements. It follows from this definition that the cohomology of a group \(G\) is detected by restricting to the subgroups \(H\) of \(G\) with \(\text{Ess}^*(G)\neq 0\). From this point of view a group-theoretic characterization of groups \(H\) with \(\text{Ess}^*(G)\neq 0\) is of considerable interest. Clearly such groups must be \(p\)-groups. In this paper the authors characterize a subclass; they show that for a \(p\)-group \(G\) one has \(\text{Ess}^*(G)\neq 0\) and \(H^*(G,\mathbb{F}_p)\) Cohen-Macaulay if and only if \(G\) has the property that every element of order \(p\) is central. Examples are given of \(p\)-groups with non-central elements of order \(p\), but with \(\text{Ess}^*(G)\neq 0\). In the last section the authors demonstrate by examples how their results can be used to compute the cohomology of certain finite groups.