I know this is more than a month after the question...I'm not on this thread much.
Yes, you can assign cohorts at the system level. That said, I would probably not use it for your need because a cohort is added to a course, it doesn't apply to all courses and I doubt that is what you want.
You could create a Teacher A , B, etc role that has the permissions you want them to have and assign the teachers to that role in categories (if you made it a category role) or at the site level. To test that it does what you want, you can either login as one or more of them and test the permissions or you could create a couple test accounts (One Teacher, Two Teacher, etc), assign them the role and then login to that account (maybe in a different browser) and test it out.
It would be helpful to know what functions you want group A teachers to have vs group B. Is it really a permissions issue or it is a groups issue?
I deal with hundreds of schools with many employees. I have a course where some administrators want all their employees to complete and want to be able to see which employees are done, etc. I'm not giving them teacher permissions. I'm not giving them site permissions. But they need to see their employees. I assign all employees to a group and I add a non-editing teacher role to that administrator and make sure he or she is assigned to the group. Now he/she sees their employees and no one else. They can't change my course. They can't change grades. They can see their staff, reports on their staff, the grades for their staff. Is this what you are looking for?