Given a group G, we can define Gop to be a new group whose underlying set is the same as that of G but with the new multiplication g.h = hg, i.e. multiply as if you were in G but reverse the order. 


Is there anything interesting to say about this construction? 


When is a group isomorphic to its opposite group?


One sees the "opposite ring" of a non-commutative ring showing up in a crucial way in number theory in the study of Brauer groups, so this is not necessarily an artificial question. This is just a question that has been poking around my brain for a little while. . . it seems like a potentially neat little problem. 



Silly Question about "opposite groups"
Every group is isomorphic to its opposite group, the map $g \mapsto g^{-1}$ being an isomorphism.


In ring theory this is not the case, and the opposite ring is an important construction. For instance, if $A$ is a central simple algebra over a field (or an Azumaya algebra over a ring), then the classes of $A$ and $A^{\operatorname{op}}$ are inverse in the Brauer group.


I don't know of anything similarly interesting to say about opposite groups.


