Difficult to know how to proceed to be honest, and I have to go out in 15 minutes, with 40 minutes worth of other "stuff" to do before I do.
⟨a,b∣ some relationships ⟩=GMeans that the group G can be generated from the two elements a,b, subject to some relationshpis on a,b.
We'll ignore the relationship part as you're dealing with generators for a specific group whose structure is known.
Note G is a group, not just a set; there is a group operation on its elements.
So, we can reduce this to
⟨a,b⟩=G for now; the format used in the question.
Notice also that these are equal, "<a,b>" is a group; it is not an
element of G.
a,b stand for two elements of the group, AND you need to replace them with actual elements of the group, e.g <i>, just a single generator, would generate the group i^0,i^1,i^-1,i^2, etc. Which works out to
1,i,−1,−iSo,
⟨i⟩={1,i,−i,−1}Now you need to find two generators, which together generate (i.e. all possible combinations of those two elements gives) the whole of
Q8. Then repeat to find all possible pairs that generate the group.