Covering numbers of unipotent conjugacy classes in simple algebraic groups (Q6620740)

The covering number cn\((G, S)\) of a subset \(S\) of a group \(G\) is the smallest integer \(k\) such that the power \(S^{k}\) is equal to \(G\), or \( \infty \) if no such \(k\) exists. Throughout the paper under review, \(G\) denotes a simple algebraic group defined over an algebraically closed field of characteristic \(p\). The author is interested in upper bounds on cn\((G, C)\) when \(C\) is a non-central conjugacy class of \(G\). It is assumed that \(p\) is a good prime for \(G\), i.e., \(p\not= 2\) if \(G\) is not of type \(A\), \(p\not= 3\) if \(G\) is an exceptional group, and \(p\not= 5\) if \(G\) is of type \(E_{8}\). This restriction on \(p\) allows the author to make use of the Bala-Carter-Pommerening classification of unipotent conjugacy classes [\textit{R. Carter}, Finite groups of Lie type: conjugacy classes and complex characters. New York: Wiley (1985; Zbl 0567.20023),Theorem 5.9.6 and \(\S\)5.11]. This classification reduces the study of unipotent conjugacy classes to the study of distinguished conjugacy classes (a unipotent element \(u\) of a group \(G\) is called distinguished if \(C_{G}(u)^{\circ}\) is unipotent). The rank \(\operatorname{rk}(H )\) of an algebraic group \(H\) is the dimension of a maximal torus of \(H\). If \(C\) is the conjugacy class of a unipotent element \( u\in G\), then the corank \(\operatorname{crk}(C)\) of \(C\) is defined as \(\operatorname{crk}(C) = \operatorname{rk}(C_{G}(u))\), and the rank \(\operatorname{rk}(C)\) of \(C\) is defined to be \(\operatorname{rk}(C) = \operatorname{rk}(G) - \operatorname{crk}(C)\).\N\NUsing this notation, the main results of the paper (Theorems A, B and C) can be exposed as follows.\N\NTheorem A. There is a constant \(c\) such that for any simple algebraic group \(G\) over an algebraically closed field of good characteristic and any distinguished unipotent conjugacy class \(C\) of \(G\) we have \(\operatorname{cn}(G, C) \le c\). Moreover, we may choose \(c = 2^{3} \cdot 3^{2}\).\N\NTheorem B. There is a constant \(c\) such that for any simple algebraic group \(G\) over an algebraically closed field of good characteristic and any unipotent conjugacy class \(C\) of \(G\) we have\N\[\N\mathrm{cn}(G, C) \le c \cdot \frac{\operatorname{rk}(G)}{\operatorname{rk}(C)} = c \cdot \left( 1 + \frac{\operatorname{crk}(C)}{\operatorname{rk}(C)} \right).\N\]\NMoreover, we may choose \(c = 2^{5} \cdot 3^{2}\).\N\NTheorem C. There is a constant \(c\) such that for any simple algebraic group \(G\) over an algebraically closed field of good characteristic and any unipotent conjugacy class \(C\) of \(G\) we have\N\[\N\operatorname{cn}(G, C) \le c \cdot \frac{\dim(G) }{\dim(C)}.\N\]\NMoreover, we may choose \(c = 2^{9} \cdot 3^{2}\).