@ChrisDavis Maybe others in the community have other ideas but I think you will need to use sections and not groups. Groups are meant for group work, so I am not aware of any way to hide the other members of the groups from them. When you are added to a group, Canvas even gives you a "Groups" button in the Global Navigation menu to easily interact with your group members.
With sections, on the other hand, students do not get any tools to see who the other members of that section are and if you hide the "People" button on the course menu, students should not be able to see any other students in the course at all (unless you have them interacting in something like a discussion).
Here are a few guides that will help with sections:
How do I add a section to a course as an instructo... - Instructure Community
How do I assign an assignment to a course section? - Instructure Community
Again, maybe others have ideas on how to make groups work the way you are wanting.
Hope this helps!
-Nick
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