Covering the set of \(p\)-elements in finite groups by proper subgroups (Q6650520)

Let \(G\) be a group, \(S\) a subset of \(G\) and \(\mathcal{H}=\{H_{1}, \ldots, H_{n} \}\) a set of subgroups of \(G\). Then \(\mathcal{H}\) cover \(S\) if \(S \subseteq \bigcup_{i=1}^{n} H_{i}\). If \(G\) is finite let \(p \in \pi(G)\) and let \(G_{p}\) be the set of \(p\)-elements of \(G\).\N\NThe aim of the paper under review is to study the minimal size \(\sigma_{p}(G)\) of a covering of \(G_{p}\) by arbitrary proper subgroups of \(G\). The authors show that if \(G=\langle G_{p} \rangle\) is solvable and not a \(p\)-group, then \(\sigma_{p}(G)\) is equal to 1 less than the minimal number of proper subgroups of \(G\) whose union is \(G\) and, for \(p\)-solvable groups \(G=\langle G_{p} \rangle\), \(\sigma_{p}(G) \geq p+1\). They also prove that if \(p\) is a fixed prime, then \(\sigma_{p}(A_{n}) \rightarrow \infty\) as \(n \rightarrow \infty\).