The tensor product, G = AxB of groups A,B satisfies:
* g in G = (a,b) forall a in A, b in B
multiplication on G is defined pairwise:
forall g1,g2 in G: g1*g2=(a1,b1)*(a2,b2)=(a1*a2,b1*b2)
identity of G = (ea,eb) for ea the identity of A and eb the identity of B.
A' subset of G: A'== A x eb ~= A... Similary with B.
A', B' normal w.r.t. G.
G/A'~=B
G/B'~=A
Given a group G, G is a tensor product of 2 smaller groups if you can write:
g=a'b'=b'a'
Example:
C6=C2xC3
Converse is not true:
let H normal subset of G
G/H=G' !=> G~=HxG'
Example: C2xC2 ~=D2 but C4/C2~=C2