This question probably has a simple and immediate answer which escapes me now. (And, I should admit, it's more my curiosity than anything else.) The only natural way to construct a group structure on the cartesian product $G\times H$ of two groups $G$ and $H$ (in particular, ``natural'' to me means that on each factor the group product should be the original one) is the semi-direct product in the case when one group acts on another one by automorphisms. Are there any natural constructions of a group structure on $G\times H$ where neither factor is a normal subgroup?


Update: I was pointed out that the notion of Zappa-Szep product that appears in the answer given by Steven Gubkin is also mentioned in [an earlier MO discussion](https://mathoverflow.net/questions/5528/when-does-a-subgroup-h-of-a-group-g-have-a-complement-in-g); I thought I'd link it here for some sort of connectivity.



Group structures on the cartesian product of two groups
Wikipedia to the rescue!


<http://en.wikipedia.org/wiki/Zappa-Szep_product>


