Elements of finite simple groups have short normal subsets representations

You need to know: Group, finite group, subgroup, identity element e of a group, logarithm $\log$ .

Background: A subset 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. For any finite set T, denote $|T|$ the number of elements in it.

The Theorem: On 28th October 1999, Martin Liebeck and Aner Shalev submitted to Annals of Mathematics a paper in which they proved, among other results, that there exists a constant c such that if G is a finite simple group and $S\neq \{e\}$ is a normal subset of G, then, for any $m \geq c \log |G|/\log |S|$ , any element of G can be expressed as a product of m elements of S.

Short context: Because there are $|S|^m$ ways to write a product of m elements of S, the conclusion in the Theorem cannot hold if $|S|^m < |G|$ , or $m< \log |G|/\log |S|$ . Hence, the inequality $m \geq c \log |G|/\log |S|$ in the Theorem is the best possible up to constant factor. The Theorem has many applications, see here for an example.