On the conjectures of J. Thompson and O. Ore

Abstract

Abstract. If G is a finite simple group of Lie type over a field containing more than 8 elements (for twisted groups l Xn(q l ) we require q > 8, except for 2 B 2 (q 2 ), 2 G 2 (q 2 ), and 2 F 4 (q 2 ), where we assume q 2 > 8), then G is the square of some conjugacy class and consequently every element in G is a commutator.

The Waring problem for finite simple groups

“…This proves a longstanding conjecture of Ore. There were many partial results on this conjecture; most notably, Ellers and Gordeev [EG98] had used entirely different methods to prove the result for all the Chevalley groups over fields of size at least 8 (and in many cases even smaller).…”

“…Let C i be the conjugacy class of s i in Γ. A theorem of Ellers and Gordeev (see for instance Theorem 1 in [EG98] and the references therein) implies that C 1 C 2 ⊇ Γ \ {1}. The result follows.…”

“…Results of [EG98], [MSW94], and [LM99] imply Theorem 1.1.4 for all but one infinite family of finite simple groups. Here we handle this remaining case.…”

Word maps, conjugacy classes, and a noncommutative Waring-type theorem

“…For some particular classes of regular semisimple elements, namely those in split tori, it has been shown by Ellers and Gordeev that C 1 C 2 Ã G n f1g (see for example Theorem 1 in [5]). However, the behavior of C 1 C 2 for classes C i of regular semisimple elements outside split tori is much less understood (see Gow [11] for interesting partial information).…”

“…Both conjectures are known to be true for alternating groups, and for groups of Lie type over fields with more than 8 elements [5].…”