Summary: Journal of Mathematical Sciences, Vol. 107, No. 5, 2001
ON APPROXIMATION OF GROUPS, GROUP ACTIONS, AND
M. A. Alekseev, L. Yu. Glebskii, and E. I. Gordon UDC 512.54
We give new examples and criteria in the theory of approximation of groups by finite groups.
Bibliography: 17 titles.
In , a class of groups locally embeddable into the class of finite groups (LEF-groups)
was introduced and thoroughly investigated. This class arises in a natural way in various
approximation problems. First of all, the definition of local embeddability into the class of
finite groups is a particular case (for groups with the discrete topology) of the definition of
approximability of topological groups by finite ones that was introduced and investigated by
the third author in connection with the problems of approximation of operators in function
spaces on topological groups. In particular, it was this definition that allowed him to study the
convergence of the approximation of the Fourier transform in the Hilbert space of functions
on a locally compact group by a finite Fourier transform  and to construct and investigate
approximations of operators in the function spaces on such groups. These approximations were
based on the discretization of their symbols .
In A. M. Vershik's paper , the notion of "closeness" of infinite-dimensional groups to
finite ones is understood as the possibility of approximating their group algebras by finite-