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 = (G^{ab})⊗_{Z}
(H^{a 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^{-1}h^{-1}

This map is not bimultiplicative but instead satisfies

[gg',h] = [^{g}g',^{g}h][g,h],

[g,hh'] = [g,h][^{h}g,^{h}h'],

where ^{h}g = 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) = (^{g}g'⊗^{g}h)(g ⊗h)

(g ⊗hh') = (g ⊗h)(^{h}g ⊗^{h}h'
)

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 H_{2}(G).