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An example is given of a 3-generator, 3-relator group whose derived group is a free abelian group of infinite rank. 1. Very little is known about the normal subgroup structure of finitely presented groups. Indeed it seems to have been fairly widely believed (since 1959, if not before) that an abelian normal subgroup A of a finitely presented group is of finite rank.2 The purpose of this note is to put paid to this belief by exhibiting a 3-generator, 3-relator metabelian group G whose derived group G' is a free abelian group of infinite rank. THEOREM. The group G = (a, s, t; st= s, [at, a] 1, as aal) is a metabelian group w,hose derived group is free abelian of infinite rank. 2. We begin by proving that G is metabelian. It follows from the defining relations of G that the derived group G' of G is generated by the conjugates at's' (i, j arbitrary integers). So it suffices to prove that these conjugates commute. The crux of the proof is the verification that the conjugates at' (i=0, ? 1, ?2, ) commute. Now gp(a, at) is abelian. Suppose that we have proved that gp(a, at, , atn) is abelian for some n? 1. Then gp(at, at2 . . . at+') is also abelian. Hence a' commutes with a, at, atn+1 and a" commutes with a, at, atn+. Bearing this in mind, and noting the defining relations of G, we find 1-[a, atn]s = [as, a tnI] = [as, astn] = [aat, (aat)t'] = [aat, atnat+'] = [a, atn+1] So it follows that gp(a, at, * * , at1) is also abelian. But this implies that (1) [a, att] = 1 (i 0, ?1, 42, Received by the editors April 20, 1971. AMS 1970 subject classqifcations. Primary 20F05, 20E15.

A finitely presented metabelian group with a free abelian derived group of infinite rank