If N is a product of two prime powers, map (x,y)->x^Ny^N is surjective on every finite non-abelian simple group

You need to know: Prime number, group, identity element, finite group, notation $g^{-1}$ for the inverse of group element $g$ , subgroup.

Background: A subgroup S of group G is called normal, if $g\cdot a\cdot g^{-1}\in S$ for any $a \in S$ and $g \in G$ . A group G is called simple if it does not have any normal subgroup, except of itself and $\{e\}$ , where $\{e\}$ is the set consisting on identity element only. A group G is called abelian if $g\cdot h = h \cdot g$ for every $g \in G$ and $h \in G$ , and non-abelian otherwise.

The Theorem: On 1st May 2015, Robert Guralnick, Martin Liebeck, Eamon O’Brien, Aner Shalev, and Pham Tiep submitted to Inventiones Mathematicae a paper in which they proved the following result. Let p, q be prime numbers, let a, b be non-negative integers, and let $N=p^aq^b$ . Then every element g of every finite non-abelian simple group G can be written as $g=x^Ny^N$ for some $x,y \in G$ .

Short context: Given group G, let w be a map mapping pair $x,y \in G$ into $w(x,y)=x^Ny^N$ . Let $w(G)$ be the set of all $g\in G$ such that $g=w(x,y)$ for some $x,y\in G$ . Map w is called surjective on G if $w(G)=G$ . In this language, the Theorem states that w is surjective on every finite non-abelian simple group G. Earlier, this result was proved for commutator map $w(x,y)=x^{-1}y^{-1}xy$ , see here. Also, it is known that $w(G)\cdot w(G)=G$ for much broader class of maps w, see here.