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1. Varieties of groups. The study of varieties of groups (more generally, varieties of algebras) has its roots in the work of Birkhoff [4] and Neumann [60] in the 1930’s. A variety of groups [61] is the class of all groups satisfying a certain family of laws; for example, the class of all abelian groups of exponent (dividing) n satisfies the laws fly—lazy= 1 and m"= 1. From this definition it is immediately clear that there are no more than 2 “0 varieties of groups. But it was not until 1970 that 01’sanskii [63] showed that there are precisely 2" 0 varieties of groups. We will see that this is very different from the situation for varieties of topological groups. Birkhoff observed that there is an alternative, but equivalent, definition for varieties of groups. A variety of groups can be described as a class of groups closed under the operations Q, S, and 0, where Q denotes a quotient group, S a subgroup, and C an arbitrary Cartesian product.(To be …