TITLE: A question about Non-Abelian Tensor Product
QUESTION [5 upvotes]: Recently, I read an old paper "Some computations of Non-Abelian Tensor Products of Groups" By R. Brown, D.L Johnson and E.F Robertson. There are some assumptions to construct this structure. They pointed that :

"Let $G$ and $H$ be groups which acts on themselves by conjugation and each of which acts upon the other in such a way that the compatibility conditions hold."

What is the action in which two groups acts on each other? I see the relations in a tensor product is the same as for commutators. Please send me a useful link about this structure. Thanks

REPLY [4 votes]: The trick is to expand $mm' \otimes nn'$ in two ways to obtain the equation
$$ ^{mn}(m' \otimes n')(m \otimes n)= (m \otimes n) \; ^{mn} (m' \otimes n'). $$
For more details, see the 1987  paper by Brown and Loday in Topology from my publication list. 
For more recent papers search the arXiv for "nonabelian tensor product" and see the bibliography 
http://groupoids.org.uk/nonabtens.html
Edit: May 16,2013  I realise I need to make one comment on Arturo's answer, namely that when we say $G,H$ act on each other that means as groups, not just as sets. A very good example of this is when $G,H$ are both normal subgroups of a group $P$ say, so that $G$ acts on $H$ via $P$ and similarly for $H$ on $G$. More generally,  this happens if $G,H$ are crossed $P$-modules. 
Another example is when two copies of the additive group of integers $Z$ act on the other one by $n \mapsto -n$ and on itself trivially. The paper by Gilbert and Higgins on the list calculates the nonabelian tensor product as $Z \times Z$. 
It should be helpful to look at arXiv:1106.2136 for generalisations.