On minimal coverings and pairwise generation of some primitive groups of wreath product type (Q6608217)

The covering number \(\sigma(G)\) of a non-cyclic finite group \(G\) is the smallest positive integer \(k\) such that \(G\) is a union of \(k\) proper subgroups. Let \(G\) be a finite group that can be generated by \(2\) elements. The generating graph of \(G\) is the simple graph whose vertices are the elements of \(G\) and two vertices are connected by an edge if they generate \(G\). The clique number of the generating graph of \(G\) is denoted by \(\omega(G)\). It is easy to see that \(\omega(G) \leq \sigma(G)\). There are many papers discussing these two invariants.\N\NLet \(A_n\) and \(S_n\) be the alternating and symmetric groups of degree \(n \geq 5\) acting on \(\{ 1, \ldots , n \}\). Let \(\tau\) be the transposition \((1,2)\). Let \(m\) be a positive integer. Let \(\delta\) be the permutation \((1, \ldots, m)\) and let \(\gamma\) be \((1, \ldots, 1, \tau) \delta \in S_{n} \wr S_{m}\). For elements \(x_{1}, \ldots , x_{m} \in A_n\), we have\N\[\N(x_{1}, \ldots , x_{m})^{\gamma} = (x_{m}^{\tau}, x_{1}, \ldots , x_{m-1}) \in {(A_{n})}^{m}.\N\]\NFinally, let \(G = G_{n,m}\) be the semidirect product \({(A_{n})}^{m} : \langle \gamma \rangle\).\N\NFor a positive integer \(x\), let \(\alpha(x)\) be the number of distinct prime factors of \(x\). The first main theorem of the paper is the following:\N\NTheorem 1. Let \(G = G_{n,m}\) for \(n \geq 30\) divisible by \(6\) and \(m \geq 2\). Then \N\[\N\sigma(G) = \alpha(2m) + {\Big( \frac{1}{2} \binom{n}{n/2} \Big)}^{m} + \sum_{i=1}^{(n/3)-1} \binom{n}{i}^{m}.\N\]\NMoreover, \(G\) has a unique minimal covering consisiting of maximal subgroups.\N\NFor \(m=1\) we have \(G_{n,1} = S_n\) and thus Theorem 1 generalizes the main result of \textit{E. Swartz} [Discrete Math. 339, No. 11, 2593--2604 (2016; Zbl 1344.20036)].\N\NThe second main theorem of the paper is asymptotic.\N\NTheorem 2. Let \(G = G_{n,m}\) with \(n\) even and \(m \geq 2\). As \(n\) goes to infinity, \(\omega(G)\) is asymptotically equal to \({( \frac{1}{2} \binom{n}{n/2} )}^{m}\) and \(\omega(G)/\sigma(G)\) tends to \(1\).