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.

Let F be a free group of rank d ≥ 2. Consider the quotient F / [ γ 4 (F ′) , F ] F ″ ′ , 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 complete description of an elementary abelian 2-group of rank ( d 4 ) in F / [ γ 4 (F ′) , F ] F ″ ′ in terms of generators of F for d ≥ 4. Our approach makes essential use of homological methods. [ABSTRACT FROM AUTHOR]