In Brave New World, how do Solidarity groups help to achieve the World State motto?
The motto of the World State is "Community, Identity, Stability." Solidarity Service groups help to achieve all of these three goals.
Every other week, people have to go to Solidarity Services with the other members of their groups. At these services, they take soma, participate in some chants, and have an orgy. These services are meant to make them feel more part of a community. By going to these services, they share a common experience with the other members of their group. That makes them feel more like they belong to that community and it gives them an identity (as part of the group). This promotes stability by making them feel more belonging.
In addition, the services offer emotional release. The drugs, the chanting and the sex are extremely powerful experiences and leave most of the participants drained and cleansed and happy. By doing so, the services can make the people more contented and less likely to rebel.