Jos Gibbons, it's not only your choice of words, I find it difficult to understand your writings because of the formalistic style also. Suppose you are right and I'm wrong; if I don't understand what you say I will never realize it.

because being unlike the rest of the group is sometimes more successful than being like them

Yes, and this is why the diversity appears in the first place. Again, I do not claim that diversity and equilibria appears in the first place because of group selection. I claim that those make group selection possible.

Competition on the genetic level was necessary for the emergence of a genetic cooperation, absolutely. But once this cooperation was established, and the genes were grouped together in a stable entity (the cell), they replicated together, not individually. The fitness of the individual gene became dependent on the other genes present.

group selection is when the reason a property has become prevalent is because of how much more successful it makes a group in which it is prevalent than one in which it is not rather than how much more successful it makes an individual with that property than one without it.

And the gene clearly makes its "group", the genome in the cell, more sucessful. It cannot reproduce on its own. Therefore, it makes sense to regard the genome level as a new level where selection takes place. Again, doesn't mean that the genetic level disappeared.

Just as genes needed a cell in order to form a new level of competition, individuals need some way of forming stable groups in order to form a new level of competition. I suggested that humans may be special in this regard, because we have things like culture and religion that binds groups together over time.

So why are you then going on about examples I never mentioned? Mice don't have culture and don't form groups that are stable over time. But humans do.