ABSTRACT
In this chapter we show how concepts such as an abelian group, the commutator of a group and related concepts like solvable groups may be generalized to arbitrary universal algebras. The commutator of two elements a and b in a group g is the element [a, b] := a-lb-lab. The commutator group of Q is the normal subgroup of g which is generated by the set {[a, b] I a, b E G} of all commutators of g. In a sense, the commutator subgroup of a group g measures how far the group is from being commutative. More generally, if M and N are two normal subgroups of g then the commutator group of M and Al, written as [.M, N], is the normal subgroup of g generated by the set {[a, I a E M,b E N}. It is well known that [JV1,.Ar is the least normal subgroup of g with the property that gh = hg for all g E MI[m„All,h E
We begin by generalizing the concept of an abelian group to arbitrary algebras. If g is an abelian group then the n-ary term operations of g are exactly operations induced by the terms t(xl , , xn) = xki • •• •• xnkn, for
, k„ E Z. These terms are normal forms for arbitrary n-ary terms over the variety of all abelian groups, meaning that any n-ary term is equivalent to one of these terms modulo IdV, where V is the variety of all abelian groups. We saw in Section 11.4 that a group g is abelian if its term operations satisfy the so-called term condition (TC) of Definition 11.4.10: for any n-ary term t (with induced term operation tG) and any elements a, b, c2, • • • ,
d2, , dn E G
(TC) t° (a, c2, , cn ) = tg(a,d2,...,dn ) t° (b, c2 , . • • , Cn) = tg (b, d2, • • • , dn) •
This gives us an equivalent characterization of the abelian property for groups, by means of term operations, and suggests how to generalize the concept of abelian to arbitrary algebras.
