Non - Hopfian Groups with Fully Invariant Kernels . I

Let Jl consist of the groups GÍL m) = (a, b; a~ b a = b"1) where |/.| 4 1 4 |zzi|, lm40 and l, m are coprime. We characterize the endomorphisms of these groups, compute the centralizers of special elements and show that the endomorphism a —• a, b —' b is onto with a nontrivial fully invariant kernel. Hence G(l, m) is non-Hopfian in the'fully invariant sense.' Our purpose is to prove the results announced in [l], concerning the endomorphisms of the non-Hopfian one-relator groups X found in [2l and isolated by G. Baumslag in [3]. Aconsists of the groups G(l, m) presented by (a, b; a~lb a = bm) where |/| 4 1 4 \m\> lm 4 0 and /, 772 are coprime. Let G' denote the normal closure of b in G (I, m) and N the kernel of the endomorphism 77: a —> a, b —* b . We will prove Theorem 1. If r: a—► A, b—► B 4 1 defines an endomorphism of G (I, m) then (1) B zs ¿72 G' andean be written in the form B = D~ b D (D in G(l, m)). (2) DAD~ = ca where c is in the centralizer of b in G' ■ Theorem 2. N is a proper fully invariant subgroup of Gil, m) such that G (I, m)/N is isomorphic to Gil, m). A general reference for the proofs of these theorems is [4]. 1. Basic lemmas. Lemma 1. 77: a—► a, b —► b defines an onto endomorphism of G(/, 272) with nontrivial kernel N where N is the normal closure of the subgroup generated by W(a, b) = ([b, aYb^b'1 and Via, b) = a-lbai[b, a]'bs)-m such that (m l)t + Is = 1. Received by the editors July 15, 1971. AMS 1970 subject classifications. Primary 20F05; Secondary 20E05.