Identical relations in simple groups - Oates, S (1963)
An identical relation of a group G is an equation of the form w(x1, …, xn) = 1, where w is an element of the free group generated by x1, …, xn, which is satisfied by any substitution of elements of G for the variables x1, …, xn. A consequence of a given set of identical relations W is a relation which holds on every group on which each member of W holds. A set of identical relations is said to be closed if it contains all its consequences. As a special case of a theorem of G. Birkhoff, contained in his paper "On the Structure of Abstract Algebras", Proc. Cam. Phil. Soc. 31 (1935) 433-454, we have that there is a one-one correspondence between varieties of groups and closed sets of identical relations. A set of identical relations is said to be finitely based if it is the set of consequences of a finite subset (which can be taken to consist of just one element, the direct product of the elements of this subset). In his paper "Identical Relations in Groups, I", Math. Ann. 14 (1937) 506-525, B. H. Neumann considers the question of whether the identical relations of a given variety (and, in particular, the variety generated by a finite group) are finitely based. He shows this to be true for a variety of abelian groups, and R. C. Lyndon, in "Two Notes on Nilpotent Groups", Proc. Amer. Math. Soc. 3 (1952) 579-583, extends this to nilpotent groups. [...] [This is the start of the abstract. For the full abstract, please consult the PDF.]