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One of the main goals of these notes is to explain how rotations in reals^nare induced by the action of a certain group, Spin(n), on reals^n, in a waythat generalizes the action of the unit complex numbers, U(1), on reals^2, andthe action of the unit quaternions, SU(2), on reals^3 (i.e., the action isdefined in terms of multiplication in a larger algebra containing both thegroup Spin(n) and reals^n). The group Spin(n), called a spinor group, isdefined as a certain subgroup of units of an algebra, Cl_n, the Cliffordalgebra associated with reals^n.  Since the spinor groups are certain well chosen subgroups of units ofClifford algebras, it is necessary to investigate Clifford algebras to get afirm understanding of spinor groups. These notes provide a tutorial on Cliffordalgebra and the groups Spin and Pin, including a study of the structure of theClifford algebra Cl_{p, q} associated with a nondegenerate symmetric bilinearform of signature (p, q) and culminating in the beautiful "8-periodicitytheorem" of Elie Cartan and Raoul Bott (with proofs).

Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions