I have been reading the paper "Point Groups and Space Groups in Geometric Algebra" by David Hestenes as part of my introduction seminar to GA. On page 5 the remark is made, that to prove that $C_p$ and $H_p$ are the only point groups in 2D, one would need to prove the following: Consider the possibility of a group generated by three distinct vectors a, b, c in the same plane. If they are to be generators of a symmetry group, then each pair of them must be related by a dicyclic condition like ($(ab)^p= −1$). It can be proved, then that one of the vectors can be generated from the other two, so two vectors suffice to generate any symmetry group in two dimensions.
Conceptually this proof makes a lot of sense to me, since the dicyclic condition implies that a rotation of 180° can be made with only two vectors, therefore creating any vector along its path in both directions, meaning any vector at all in two dimensions. But trying to prove this in geometric algebra, I always found myself relying on linear algebra and my understanding of basis to make this rather basic proof. And it made me question my understanding of the rest of the paper, since it is never really defined which operation is even being used.
My idea to solve this with GA would be to use (ab) to define a rotation from a point x as it was done above in the paper and from there try to show that with a precise rotation, a general c can always be found. But I am unsure how the condition of c having to be related to the other vectors with the dicyclic condition would come into play.
Could someone please explain how this would be done? And I apologize for the rather trivial question.