On elements of order 2 in the free centre-by-(nilpotent of class 3)-by-abelian group within the variety of all soluble groups of derived length 3

Abstract

Let F be a free group of rank $d\ge 2.$ Consider the quotient $F/[{\gamma }_4(F^{\prime }),F]F^{″\prime },$ the free centre-by-(nilpotent of class 3)-by-abelian group within the variety of all soluble groups of derived length 3. A peculiar feature of this group is the occurence of torsion in its centre. Our main result is a description of an elementary abelian 2-group of rank $\left(\begin{array}{c}d\\ 4\end{array}\right)$ in $F/[{\gamma }_4(F^{\prime }),F]F^{″\prime }$ in terms of generators of F for $d\ge 4.$ Our approach makes essential use of homological methods.

Keywords:

• Free central extensions of groups
• Homology of groups

2020 Mathematics Subject Classification:

• 20E22
• 20J05

Acknowledgments

The author would like to thank the anonymous referee for helpful comments and suggestions that improved the presentation of the paper.