Andrews-Curtis conjecture

From Groupprops
Jump to: navigation, search
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.

External links