This is the paper that is treated as a gentle introduction for mathematicians to the algebra of the Standard Model of particle physics which is actually the best tested and the most widely accepted theory of describing all the particles and all the forces of nature, except gravity. Three grand unified theories which unify particles and forces beyond the Standard Model via extending the latter gauge group U(1) $\times$ SU(2) $\times$ SU(3) are thoroughly reviewed. These are Georgi-Glashow's SU(5) theory, Georgi's theory based on the Spin(10) or SO(10) group, and the Pati-Salam model based on the SU(2) $\times$ SU(2) $\times$ SU(4) group.