# Group structures on the cartesian product of two groups

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; I thought I'd link it here for some sort of connectivity.

Wikipedia to the rescue!

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

• Dear Steven, thanks! That (more precisely, the part on external Zappa-Szep products) is exactly what I was looking for. – Vladimir Dotsenko Jun 13 '10 at 19:37
• Interstingly enough, this construction -- which I knew but not by name -- is intimately related to the subject of "dressing transformations" in integrable systems. The actions of H on K and of K on H are abstract analogues of the dressing actions in Poisson-Lie groups. – José Figueroa-O'Farrill Jun 13 '10 at 20:30
• @Jose: Yes, and also there is an analogous version for Hopf algebras. Indeed, the "Drinfeld Double" of a Hopf algebra is a special example. – Theo Johnson-Freyd Jun 13 '10 at 22:01
• As José and Theo remarked, this is certainly an interesting notion, but I don't see any general (non-tautological) $\textit{construction}.$ – Victor Protsak Jun 13 '10 at 23:54
• @VladimirDotsenko: The relationship with semi-direct product is closer than your last remark hints at: $\alpha$ must be a left action of $K$ on $H$, and $\beta$ must be a right action of $H$ on $K$. (And then those two actions must satisfy some compatibility assumption.) And indeed, if either of these actions is trivial, then one recovers the semi-direct product. – Joshua Grochow Mar 20 '17 at 15:19