Andrews-Curtis conjecture

Statement
Let $$F_n$$ denote the free group on $$n$$ elements with a set $$X = \{ x_1,x_2,\ldots,x_n \}$$ a freely generating set. Then, the following holds:

A set $$Y = \{ y_1,y_2,\ldots,y_n \}$$ of elements of $$F_n$$ generates $$F_n$$ as a normal subgroup if and only if $$Y$$ is Andrews-Curtis equivalent to $$X$$, viz one can get from $$X$$ to $$Y$$ by a sequence of Nielsen transformations along with inner automorphisms from $$F_n$$.

There is also a stronger version, the stable Andrews-Curtis conjecture.