Normal \(2\)-coverings of the finite simple groups and their generalizations (Q6553922)

Let \(G\) be a finite group and let \(H \leq G\). An easy counting argument shows that \(\bigcup_{g \in G} H^{G} \not =G\). However, there are examples of finite groups which are the union of conjugates of two proper subgroups (typical examples are Frobenius groups).\N\NA normal (weak normal) \(k\)-covering of \(G\) is a set \(\mu= \{H_{1}, \ldots, H_{k} \}\) of \(k\) proper subgroups of \(G\) with the property that every element of \(G\) belongs to the conjugate \(H_{i}^{g}\), for some \(i \in \{1, \ldots , k\}\) and for some \(g \in G\) (\(g \in \Aut(G)\)). The normal (weak normal) covering number of \(G\), denoted by \(\gamma(G)\) (\(\gamma_{w}(G)\)), is the smallest integer \(k\) such that \(G\) admits a normal (weak normal) \(k\)-covering.\N\NThe purpose of this monograph is the classification of finite non-abelian simple groups such that \(\gamma_{w}(G)=2\) and \(\gamma(G)=2\) (an effective list appears in Tables 1.3, 1.4, 1.5, 1.6 and 1.7).\N\NIn the introduction, the authors give various motivations for their investigation. Besides the intrinsic theoretical interest in (weak) normal \(2\)-coverings, a few remarkable applications have arisen: ranging from Galois theory, the generation of simple groups and the density of derangement graphs. Each of these applications has a number of interesting open questions and conjectures, which might benefit from having a well-established theory of weak normal coverings. Therefore, it is important to collect all partial results already available in the literature and to have a comprehensive analysis of the most basic case, that is, weak normal 2-coverings of non-abelian simple groups.\N\NIt is interesting to note that there are similarities between the maximal components of \(2\)-coverings (listed in the main theorem) and the non-trivial maximal factorizations of simple groups (which were classified by \textit{M. W. Liebeck} et al. [The maximal factorizations of the finite simple groups and their automorphism groups. Providence, RI: American Mathematical Society (AMS) (1990; Zbl 0703.20021); J. Algebra 185, No. 2, 409--419 (1996; Zbl 0862.20016)]. The two categories are quite distinct, though. Simple groups, such as the symplectic groups \(\mathrm{PSp}_{6}(3^{f})\) with \(f \geq 2\), admit normal \(2\)-coverings, but lack non-trivial factorizations. Conversely, other simple groups, such as alternating groups of prime degree \(p \geq 11\), have non-trivial factorizations, but lack normal \(2\)-coverings. Hence, there exists no direct correlation between normal \(2\)-coverings and factorizations.\N\NThe book is divided into twelve chapters, the titles of which are as follows. 1 Introduction (invariably generating graph and the \(\Aut\)-invariably generating graph, the Erdős-Ko-Rado theorem and the derangement graph, normal coverings and Kronecker classes); 2 Preliminaries (classical groups, Huppert's theorem and Singer cycles, primitive prime divisors, Bertrand elements, spinor norm); 3 Linear groups; 4 Unitary groups; 5 Symplectic groups; 6 Odd dimensional orthogonal groups; 7 Orthogonal groups with Witt defect 1; 8 Orthogonal groups with Witt defect 0; 9 Proofs of the main theorems; 10 Almost simple groups having socle a sporadic simple group; 11 Dropping the maximality; 12 Degenerate normal \(2\)-coverings.\N\NThe reviewer believes that the presence of a subject index and/or a list of symbols and notations would have greatly aided the scholar in reading this interesting work.