w_1(G)w_2(G)=G for any nontrivial group words w_1,w_2 and any large finite non-abelian simple group G

You need to know: Group, identity element, finite group, notation $|G|$ for the number of elements in a finite group G, abelian and non-abelian groups, simple group, free group.

Background: Let $w = w(x_1,\dots,x_d)$ be a non-trivial group word, that is, a non-identity element of the free group $F_d$ on $x_1,\dots,x_d$ . For a group G, denote $w(G)$ the set of all elements $g \in G$ which can be obtained by substitution of some $g_1, g_2, \dots, g_d \in G$ into w instead of $x_1, x_2, \dots, x_d$ , respectively, and performing the group operation. For subsets $A,B$ of group G, let $A\cdot B=\{g \in G: g=a\cdot b, \, a\in A, \, b\in B\}$ . Denote $A^2=A \cdot A$ .

The Theorem: On 10th February 2010, Michael Larsen, Aner Shalev, and Pham Tiep submitted to the Annals of Mathematics a paper in which they proved that for each pair of non-trivial words $w_1$ , $w_2$ there exists $N =N(w_1, w_2)$ such that for every finite non-abelian simple group G with $|G|\geq N$ we have $w_1(G)\cdot w_2(G) = G$ .

Short context: In 2001, Liebeck and Shalev deduced from this theorem that for any non-trivial group word w, is there a constant $c=c(w)$ such that $w(G)^c=G$ for every large finite simple group G. In 2006, Shalev proved that this holds with $c=3$ , independently of w. In 2007, Larsen and Shalev proved that, for some groups (called alternating groups), one can even take $c=2$ . The Theorem states that one can in fact take $c=2$ for all w and all large G. This is clearly the best possible in general, but see here and here for the proof that $c=1$ works for some specific words w.