Various authors have obtained an eight term exact sequence in homology
from a short exact sequence of groups
the term V varying from author to author (see  and ; see also  for the simpler case where N is central in G, and  for the case where N is central and N ⊂ [G, G]). The most satisfying version of the sequence is obtained by Brown and Loday  (full details of  are in ) as a corollary to their van Kampen type theorem for squares of spaces: they give the term V as the kernel of a map G ∧ N → N from a “non-abelian exterior product” of G and N to the group N (the definition of G ∧ N, first published in , is recalled below). The two short exact sequences
where F is free, together with the fact that H2(F) = 0 and H3(F) = 0, imply isomorphisms
The isomorphism (2) is essentially the description of H2(G) proved algebraically in . As noted in , the isomorphism (3) is the analogue for H3(G) of the Hopf formula for H2(G).