Communicated by M. F. Newman It is a trivial consequence of Magnus' solution to the word problem for onerelator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there

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... a fixed group with twenty-six defining relations in which every recursively presented group is embedded. The results of the present paper are analogous for groups of the results of [2] about semigroups; however, no knowledge of [2] is required to read this paper. Let A be any finitely presented group. In view of the nature of our proposed theorem we may, without loss of generality, assume that A has two-generators-• say A = . (See for example [8] ). We begin by regarding A as a semigroup. In this role A has four generators, namely a u a 2 ,a^1 and a^1. When convenient, we shall sometimes write a 3 and a 4 for aj" 1 and a^1 respectively. Let A = (a u a 2 ,a 3 ,a 4 ; R t = 1, i = 1,2, •••,«). We begin by applying to A a construction of the type given on p. 307 of [4] . It will turn out that there is one delicate point in the argument. In order that this point will be clear, and to make the paper more easily readable, we specify the various constructions in detail.