The transition from the old algebra of equations to the modern algebra of group theory and Galois theory was pioneered in Russia by D. A. Grave, a student of P. L. Chebyshev, subsequently academician of the Ukrainian Academy of Sciences in Kiev. Many of his students, including B. N. Delone, M. F. Kravchuk, N. G. Chebotarev, A. M. Ostrovsky, and O. Yu. Shmidt became prominent mathematicians and further developed the new approach. In 1929 Shmidt became professor at Moscow University and founded the chair of higher algebra there. The article describes related changes in the university curriculum and the social-political context of the development of a new school in mathematics.