EXTENSIONS OF GROUP RETRACTIONS

In this paper a condition, which is necessary and sufficient, is determined when a retraction of a subgroup H of a torsion-free group G can be extended to a retraction of G. It is also shown that each retraction of a torsion-free abelian group can be uniquely extended to a retraction of its divisible closure.

abelian group.In this paper we consider the following question: if G is a torison-free abelian group, H is a subgroup of G, and T is a retraction of H, then when can T be extended to a retraction of G? In Theorem 5.7 we give a necessary and sufficient condition for the existence of such an extension.The key to proving this theorem is Theorem 3.2 where it is shown that if H is a normal subgroup of a group G such that G/H can be linearly ordered and T is a retraction of H such that for each g G and each finite nonempty sub- (g-lAg) -i set A of H, T g (AT)g, then T can be extended to a retraction of G.This theorem is a generalization of a corresponding theorem in the theory of lattlce-ordered groups (see [1], [7], or [I0]).
If G is a torsion-free abelian group and D is a divisible closure of G, then, in Theorem 4.7, we show that each retraction of G has a unique extension to a retraction of D. Again this theorem generalizes a well-known result in the theory of lattlce-ordered groups.The key in the proof of Theorem 4.7 is Corol- lary 4.4 where it is shown that each retraction of an abellan group satisfies condition (6).(Definitions will be given in Section 2.) In Corollary 4.8, we obtain a partial converse of Theorem 4.7.An immediate consequence of Corollary 5.8 is that each torsion-free abellan group admits an infinite number or retrac- tions.

PRELIMINARIES.
In this section we give some definitions and results from [2] that will be used in this paper.Throughout this paper, G will denote a group,written multi- plicatively and with identity I, and F(G) will denote the collection of all finite, nonempty subsets of G. Then F(G) is a join monoid, that is, F(G) is a join semilattice in which A v B A u B, F(G) is a monoid in which AB {abla A and b B}, A(B v C) AB v AC, and (A v B)C AC v BC.A homomorphism @ of F(G) into G such that {g} g for every g G will be called a retraction of G.We will denote by Ret G the collection of all retractions of G.If Ret G is nonempty, then G is said to be a retractable group.If G is a lattice-ordered group and o is given by Ao v A for each necessary and sufficient conditi.on that a partial order of a subgroup of a group may be extended to a partial order of the group.The main result of this section gives a sufficient condition for the extension of a retraction of a subgroup to a retraction of the group.The proof of Theorem 3.1 is immediate and will be omltted.THEOREM  We must show that o is well-defined.Suppose that d (gPg]r ,...,gngp-1}T. -1 Since gpgn E H, Thus dgp hg n and it follows that u is a function.Clearly, {g}o g for every g G. Let A be as above, B {Xl'''''Xm } F(G), where x-i Thus Ker T = Ker o and, by Theorem 3 i, G extends The verification of (i) is immediate from Theorem 3.1 and [2, Theorem 4.3   (ii)], the verification of (ii) is straightforward, and (iii) is well-known from the theory of lattlce-ordered groups.COROLLARY 3.3: If G is a torsion-free abelian group, H is a proper pure subgroup of G, and T E Ret H, then T can be extended to a retraction of G in at least two ways.
PROOF: If H is a pure subgroup of G, then G/H is torslon-free.Each torsion-free abelian group can be linearly ordered [5, p. 0.4].Since H is a proper subgroup of G, G/H admits at least two distinct linear orderlngs.Hence T can be extended in at least two ways to a retraction of G.

DIVISIBLE EXTENTIONS.
The main result of this section is that each retraction of a torslon-free abelfan group can be uniquely extended to a retraction of its divisible closure.
In Theorem 4.3 we show that this identity generalizes to retractable groups.To this end, we state two lemmas.(A proof of Lemma 4.1 can be given by induction and Lemma 4.1 can be used to prove Lemma 4.2.)LEMMA 4.1: If G is a group, A {gl,...,gm } F(G) such that gig j gg for all i and j, and n .N, then We state the following three corollaries which will be used numerous times in the sequel.COROLLARY  Ret G, then G(n) is a o-subgroup of G. Thus, the largest divisible subgroup of G is a o-subgroup.
COROLLARY 4.6: Let G be an abellan group, H a subgroup of G, and n H, g g G and g H for some n be the pure closure of H in G.If o e Ret G and H is a o-subgroup of G, then so is H,.
Let G be a torslon-free abelian group and D be a divisible closure of G.  COROLLARY 4.8: Let G be a torsion-free abelian group and H be a sub- group of G.If there exists T Ret H such that T has a unique extension to G, then G/H is torsion (or equivalently, G is contained in a divisible clo- sure of H).
PROOF: Let o be the unique extension of T to a retraction of G.If H, denotes the pure closure of H in G, then by Corollary 4.6, H, is a o- subgroup of G. Hence, T extends uniquely to a retraction of H,.Thus, by If G is a torsion-free abelian group, D is a divisible closure of G, o Ret G, and T is the unique extension of o to a retraction of D, then it is easy to show that the collection of 0-o-subgroups of G is lattice isomor- phic to the 0-T-subgroups of D.

EXTENDING RETRACTIONS.
In this section we prove the main result (Theorem 5.7) of this paper, namely, if G is a torsion-free ableian group and H is a subgroup of G, then we give a necessary and sufficient condition that a retraction of H can be extended to a retraction of G.
THEOREM 5.1: Let G be an abelian group, D a divisible closure of G, K be a subgroup of D that contains G, and o Ret G. Then o can be ex- (n) tended to a retraction of K if and only if for each n N, K n G is a o-subgroup of G.Moreover, if o extends to a retraction of K, then it is unique.
PROOF: Let T be the unique extension of to a retraction of D. If O can be extended to a retraction of K, then T is the unique extension of 9 to a retraction of D. If n N, then by Corollary 4.5, K (n) is a -subgroup of K and hence, a T-subgroup of D. Therefore, K (n Conversely, suppose that K n G is a o-subgroup of G for each n N. Let {h I ,hm F(K) and {h I, hm }T a. Then there exists n N such n ,hm n} F(G).Since satisfies (6), that {h a is a T-subgroup of D and hence, o can be extended to a retraction of K.
The uniqueness is immediate from Theorem 4.7.
From the proof of Theorem 5.1, we have COROLLARY 5.2: Let G be an abel+/-an group, D be a divisible closure of G, K be a subgroup of D that contains G, and O e Ret G.If there exists such that K(n) c G and K(n) is a u-subgroup of G, then o has a unique extension to a retraction of K.
In the following considerations (up to Corollary 5.6), for obvious reasons, we use additive notation for the binary operations on groups.Let H be a sub- group of Q and define H +/-{rlr e Q and rh H for every h H}.It is easily verified that H +/is a subring of Q containing z.Moreover, it follows from the theory of abel+/-an groups of rank i that if K is a subgroup of Q con- taining H and H + {0}, then H i is a subring of K+/-.
In [2, Example 5.7] all of the retractions of Q were exhibited, namely, Ret Q {Or[r e Q}, where Or is defined by AOr (r + l)max A r min A, for all A e F(Q).The preceding corollary is not true for abel+/-an groups of rank 2, as we illustrate with the following example.EXAMPLE 5.5: Let K Z Z and let be the -retractlon of K induced by the cardinal ordering of K. Let H K + <(,-1/2)>, where <(1/2,-1/2)> denotes the cyclic subgroup of Q Q generated by (1/2,-1/2).Then 2H 2K + <(I,-i)> and (I,-I) 2H n K. Now {(0,0),(i,-i)}--(0,0) v (i,-i) (I,0) K.However, (I,0) 2H.Thus, 2H n K is not a u-subgroup of K and so by Theorem  G(nP) 0 M. Hence, {Cl,...,Cm} X {Cl,...,Cm} G (n) n C. In particular, when n i, we have shown that X Ret(G 0 C), and for n arbitrary (n > 0), G (n) 0 C is a X-subgroup of G n C. Therefore, (G n C,X) and by the maxi- mality of (M,9), G 0 C M. Consequently, M is pure in G.By Corollary 3.3, 9 can be extended to a retraction of G and hence, T can be extended to a retraction of G.
The converse is immediate by Corollary 4.5.
Z +/-By Corollary 5 3, Ret Z { Ir } {Orlr Z}.Thus, for each sub- r group H of Z and each r Ret Z, H is a u-subgroup of Z.It follows that if G is a torsion-free abelian group and g G, then each retraction of the cyclic subgroup of G generated by g can be extended to a retraction of G. (Hence, each torsion-free abelian group has at least a countably infinite number of retractions.)Combining Corollary 5.4 and Theorem 5.7, we have a stronger result than this.
COROLLARY 5.8: Let G be a torsion-free abelian group and H be a locally cyclic subgroup of G. Then every retraction of H can be extended to a re- traction of G.
.d. (re,n)' andI/nIf g e G, we write g for that unique element in D DmD DI.

7 :
n c.m (re,n)" whose n-th power is g.For n e N, define from G into D by g@n gl/n" If G is an abellan group, D is a divisible closure of G, and oRet G, then o can be uniquely extended to a retraction T of D. and {gl,...,gm } Ker } and Ker F(G) s Ker T;(ill) Ker is a convex subset of F(G) if and only if Ker T is a convex subset of F(D) (iv) Ker o is a convex subsemilattice of F(G) if and only if Ker T is a convex subsemilattice of F(D); hence o is an -retraction of F if and only if T s an E-retraction of D. The verification that T is the unique n n nN n extension of O to a retraction of D, and the remainde of the theorem is rou- tine.
Let D be a divisible closure of G and let C be the divisible closure of M in D.Then, by Theorem 4.7, 9 can be extended to a retraction of C. Let X IF(G C).Then, we assert that (G C,X) .If n N and {c I c m} F(G (n) n C), then G (n) a (G n C) G (n) C, and there exists p N such that {c p,c p} c G np) o M. Since satslfles () and m The class of retractable groups is a proper subclass of the class of torslon-free groups [2, Theorem 2.2 and Example 2.7].
2,Theorem 2.1] and o is called the retraction of G induced by the lattice-ordering of G.
If Ker o is a convex subset (resp., convex sub- semilattice) of F(G), then o is said to be a convex retraction (resp., -retraction) of G.There is a one-to-one correspondence between the lattice- orderings of G and the -retractions of G[2, ll) H is a 0-G-subgroup of G if and only if {gl,...,gn } e Ker O and It was shown in [2, Theorem 4.3 (1)] that if o Ret G and H is a normal 0-G-subgroup of G, then the mapping o* given by {Hgl,...,Hgn}O* H({g I ,gn}O) is a retraction of G/H (called the retraction of G/H induced by o).THEOREM 3.2: Let G be a group, H be a normal subgroup of G, 3.1: Let o Ret G and H be a subgroup of G. (1) If T Ret H, then o extends T if and only if Ker T c er o. (h I h H imply that {hlg I n ''"'hngn }O H. and (g-lAg) T -i T Ret H such that for every g G and every A F(H), g (AT)g.Suppose further that (G/H,<) is a linearly ordered group.Then there is a re- traction o of G that extends T.Moreover, (1) H is a 0-O-subgroup of G, o* is the retraction of G/H induced by the linear ordering < of G/H; and hence, H is a convex o-subgroup of G; (ll) T is convex if and only if o is convex; (ill) T is an E-retractlon if and only if o is an E-retractlon.PROOF: Let A {gl,...,gn} F(G) where Hg I < < < Hgp_ 1 Hgp Hg n.E H. Lt h ...,gng n }T and define Ao hg n.
We next turn our attention to the problem of extending a retraction of a subgroup H of a torison-free abelian group G to a retraction of G. THEOREM 5.7: Let G be a torsion-free abellan group, H a subgroup of G,