This is a guide to the bibliography on non abelian tensor products
of groups.
The basic idea of the non abelian tensor product is simple.
It was early realised that a direct definition of tensor
product of non abelian groups gave nothing new. Let G,H be
groups. Define G ⊗ H as the group with generators
g ⊗ h, g ∈ G, h ∈ H and relations
for all g,g' ∈ G, h,h' ∈ H.
Then expanding gg'⊗hh' in
two ways yields (after some cancellation)
(g⊗h')(g'⊗h) = (g'⊗
h)(g ⊗ h').
From this one finds easily that
G ⊗H = (Gab)⊗Z
(Ha b) .
The start of a new approach was to recognise that if one is
interested in non commutative groups then one is certainly
interested in the commutator map on a group G
[ , ] : G ×G → G.
(g,h)→
ghg-1h-1
This map is not bimultiplicative but instead satisfies
[gg',h] = [gg',gh][g,h],
[g,hh'] = [g,h][hg,hh'],
where hg = hgh-1. A map of this type we call a
biderivation. It is thus natural to consider the universal
object for biderivations. So we now define G ⊗G to be the
group with generators g⊗h, g,h ∈ G and relations
(gg'⊗h) = (gg'⊗gh)(g ⊗h)
(g ⊗hh') = (g ⊗h)(hg ⊗hh'
)
for all g,g',h,h' ∈ G. The natural map
G ×G → G ⊗G, (g,h) → g ⊗h is then the universal biderivation and
any biderivation b : G ×G → L factors uniquely to give a
morphism of groups b' : G ⊗G →L. In particular the
commutator map defines a morphism κ: G ⊗G → G whose
image of course is the commutator subgroup [G,G] of G.
There are other relations satisfied by the commutator, for example
[g,g] = 1. It is natural therefore to consider another
construction, the exterior product
G ∧G = (G ⊗G)/{g ⊗g : g ∈ G} .
Again the commutator map yields a morphism of groups
κ': G ∧G →G.
This map was introduced in essence in 1952 by Clair Miller ('The second homology group of a group;
relations among commutators'.
Proc. Amer. Math. Soc. 3, (1952). 588-595).
who proved that the kernel of κ' is isomorphic to the Schur
Multiplicator H2(G).