Andrews-Curtis conjecture

This article is about a conjecture in the following area in/related to group theory: free groups. View all conjectures and open problems

This conjecture is believed to be false

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.