Sums of Normal Endomorphisms
نویسنده
چکیده
It should be noted that the definition (introduced in [l]) of a normal endomorphism of a loop G is radically different from the usual definition for groups. Nevertheless (as shown in [2]) the two definitions are equivalent when G is a group. In particular, the present theorem generalizes one of Heerema [3]. We may add that in [2], by assuming that the loop G was Moufang, we were able to be much more explicit than at present about the properties of the ring (L, +, ■). For example, if G is merely power-associative and if d is an element of L which happens to be an endomorphism, we can answer neither of the following questions: (i) Is the complement 1—6 an endomorphism? (ii) Is 9 semi-normal in the sense of [2]? The definition of a normal endomorphism involves the concept of a purely nonabelian (p.n.a.) loop word; this will be discussed in §2. Roughly speaking, p.n.a. loop words are the analogues for loops of the higher commutator forms introduced by Philip Hall in the theory of groups, although an inductive definition of p.n.a. loop words as "higher commutator-associator forms" does not seem profitable at present. In particular, the following simple lemma, on which the proof of the theorem hinges, could be generalized to deal with the various terms of the lower central series or derived series of a loop:
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