# Andrews-Curtis conjecture

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 $F_n$.