Let \(G\) be a compact Lie group, \(X\) be a \(G\)-space and \(f:X\to X\) be self \(G\)-map. In this interesting paper the authors deal with the following two basic questions: (1) For which groups \(G\) is every self \(G\)-map essential? (2) For which \(G\)-spaces is every self \(G\)-map essential? Here the word ``essential'' \ has its usual non-equivariant meaning, that is, a map \(X\to X\) is essential if it is not homotopic (in the ordinary, non-equivariant sense) to a constant map. To answer these questions, the authors use a very useful invariant, the \(G\)-capacity of a \(G\)-space. This invariant for free involutions was introduced in \textit{M. A. Krasnosel'skij} [Topological methods in the theory of nonlinear integral equations, Int. Ser. Monogr. Pure Appl. Math. 45 (1964; Zbl 0111.30303)], and further studied by \textit{P. E. Conner} and \textit{E. E. Floyd} [Bull. Am. Math. Soc. 66, 416-441 (1960; Zbl 0106.16301)], who called it the index of the \(G\)-space. In the first named author's paper [Comment. Math. Helv. 71, No. 4, 570-593 (1996; Zbl 0873.58018)], this invariant was generalized to arbitrary group actions in order to apply it to critical point results. Recall that for a fixed-point-free \(G\)-space \(X\), the \(G\)-capacity \(k_G(X)\) is just the greatest number \(n\) such that the \(n\)-fold join \(E_nG\) of \(G\) can be mapped \(G\)-equivariantly into \(X\). The \(G\)-capacity has the following rigidity property: if \(k_G(Y)=k_G(X)<\infty\) then every \(G\)-map \(f:X\to Y\) between fixed-point-free \(G\)-spaces is essential. The authors answer the first question stated above as follows. Theorem 1. The following are equivalent: (a) \(G\) is a \(p\)-toral group, that is, \(G\) is an extension \(1\rightarrow T\rightarrow G\rightarrow P\rightarrow 1\) of a finite \(p\)-group \(P\) by a torus \(T=\mathbb S^1\times\dots\times\mathbb S^1\); (b) The capacity \(k_G(X)<\infty\) for every finite dimensional fixed-point-free \(G\)-complex \(X\) of finite orbit type; (c) every \(G\)-map \(X\to X\) on a finite dimensional fixed-point-free \(G\)-complex \(X\) of finite orbit type is (non-equivariantly) essential. A partial answer to the second question gives the following Theorem 2. Let \(V\) be a fixed-point-free unitary representation of a finite group \(G\). Then the following are equivalent: \((a)\) there exists a prime number \(p\) which divides the cardinality of every \(G\)-orbit of the unit sphere \(SV\); \((b)\) the \(G\)-capacity \(k_G(SV)<\infty\); \((c)\) every \(G\)-map \(SV\to SV\) is (non-equivariantly) essential.