Having seen many people go through abstract algebra courses, I've noticed that one factor for success is the ability to pull out examples to test ideas on (e.g. if two subgroups have the same index, are they the same? No, the Klein four-group is a quick counterexample).
Now, my example may seem trivial, but that's the point; many students who struggle with abstract algebra don't see such things as trivial, and try to prove or disprove using symbolic manipulations instead of quickly going through a list of examples.
What are the core examples of groups that students should think of when testing ideas?
Please indicate the groups and why you think they are important.emphasized text