[Cancellation theorem][1] in group theory (for direct product) says that if $B$ is a finite group and $A \times B \simeq A_1 \times B_1$ and $B \simeq B_1$ then $A \simeq A_1.$ 

Of course, if $B$ is not finite, the result is absurd, even for finitely presented groups ([Here][2] is an example by Steve)

> I wonder whether the cancellation
> theorem holds for different products
> (in finite or infinite cases), such as
> semi-direct product, free product,
> fiber product over a given group,
> Zappa-Szep product (knit product),
> Wreath product.



  


  [1]: http://www.jstor.org/pss/2317133
  [2]: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=61&t=351637