Pacific Journal of Mathematics THE CLASSIFICATION OF CERTAIN CLASSES OF TORSION FREE ABELIAN GROUPS CHARLES ESTEP MURLEY Vol. 40, No. 3 November 1972 PACIFIC JOURNAL OF MATHEMATICS Vol. 40, No. 3, 1972 THE CLASSIFICATION OF CERTAIN CLASSES OF TORSION FREE ABELIAN GROUPS C. E. MURLEY Let jy denote the class of torsion free Abelian groups of finite rank. It is shown that for Ae JZf, there is a quotient divisible subgroup QD(A) such that A/QD(A) is a reduced torsion group. Furthermore, QD(A) and AIQD(A) are unique up to quasi-isomorphism. Let & denote the subclass of Ssf of groups A such that for almost all primes p, the p-primary component of A/QD(A) is the direct sum of Tp(A) isomorphic cyclic groups where r(A) denotes the p-rank p of A. The groups in & are classified up to quasi-isomor- phism, which generalizes the Beaumont-Pierce classification of quotient divisible groups. The main results of this paper concern the subclass i? of Sf of groups A such that r(A) ^ 1 for all primes p. The p class if may be profitably treated as a generalization of the class of rank one groups in Sf'. In §4 if is characterized as a certain subclass of the class of groups in s^f whose isomorphism and quasi-isomorphism classes coin- cide and the groups in g" are classified up to isomorphism. This generalizes the well-known Baer classification of rank one groups in sf and is related to a question of L. Fuchs concerning the structure of torsion free Abelian groups which have hereditary generating systems. In §5 the endomorphisms of groups in if are studied. It is shown that every endomorphism of an indecomposable group in if is an integral multiple of an automorphism. The special groups of F. Richman play much the same role in if that the groups of non-nil type play in the class of rank one groups in J^ For example, an indecomposable group A in if is the additive subgroup of the endomor- phism ring of some group in g7 if and only if A is a special group. In the following, Π denotes the set of primes in the ring of integers Z, Z the local subring of the field of rationale Q determined v by the prime p and Z(n) the cyclic group of order n. The ring of p-adic integers and the field of p-adic numbers are denoted by Z{p) and Q(p) respectively. Let M be a torsion free module over an integral domain R with quotient field Q(R). Then the rank of M, denoted by r (M), is the Q(i2)-dimension of Q(R) ® M. If R = Z, B R then we let r(M) = r(M) and call a subgroup N full in M if r(M) — z r(N). If B is a p-primary abelian group and B[p] = {xe B\px = 0}, then the rank of B, denoted by r(J5), is the ^(p)-dimension of B[p]. The p-rank of a torsion free group B, denoted by r(A), is r(B/pB). p 647 648 C. E. MURLEY Let B and C be Abelian groups. Then B (x) C and Horn (J5, C) will mean B(& C and ΐlom(B, C). The endomorphism ring of B is Z z denoted by End (I?). We let B denote the p-primary component of p B, d(B) be the maximal divisible subgroup of B and for B torsion free, Π(B) = {pe Π\pB = B} = {p e 771 r (jB) = 0}. If H is a charac- p teristic, then the type determined by H is denoted by [H]. Let B be torsion free and x e B. Then the height of x in 1? is a charac- teristic which we denote by HB{x). The inner type of B [13], denoted by τ*(B), is the greatest lower bound in the lattice of types of the type set of B, i.e. {[HB{x)\ | 0 Φ xe B}. For H a characteristic, we let B[H] = {xe B\HB{x) ^ H}, which is a fully invariant subgroup of B. Epimorphisms [monomorphisms] are denoted by -»[>->]. Much of this paper is contained in the author's doctoral thesis supervised by Professor R. A. Beaumont at the University of Washing- ton, Seattle, Washington. The author is also indebted to Professor R. B. Warfield, Jr., for several valuable suggestions concerning the endomorphism rings of groups in i? and to the referee for suggesting many substantial improvements. 1» The local-global setting* Throughout A will denote a tor- sion free abelian group of finite rank and will be considered as a full subgroup of a fixed finite dimensional rational vector space V. Any group in Sf with the same rank as A can be imbedded in V as a full subgroup and V= Q®A. Let V(p) = Q(p)(g)V for peΠ. Then we regard the Q{p)-module V{p) as an extension of V where V{p) Π yip) _ γ f p q Lt B be a subgroup of A. Then Z B denotes or φ φ e P the Zp-submodule of V generated by B and Blp) denotes the Z{p)-sub- module of Vip) generated by B. Since V{p) is viewed as an extension of V and torsion free groups are flat, there is a natural identification between A{p) and Z{p) (x) A given by Σr^ —> Σri (x) c^ for r* e Z(p) and ae A. A similar identification occurs between Z A and Z ®A. Note { P P that r(A) = r ( )(A(p)) = r ( )(V(p)) = r (ZA). The following well- Z P Q P Zp p known local-global relations will be frequently used: (i) A(p) ΠV= Z A P (ii) ΠZ A = A = ΠAW P qeΠ qeΠ (iii) pnZA Πi = pnA = pnA{p) f) A for n ^ 0 p (iv) for B a full subgroup of A, Z A/Z B s (A/B) s A{p)/B{p). P P p Two groups B and C are quasi-isomorphic, denoted by B = C, if there are subgroups F gβ and C SC such that B' ^ C and. B\B\ CjC are groups of bounded order. For B and C torsion, B ~ C if and only if B έ C for all p and JB = C for almost all p [4]. p p P p Thus, if B and C are torsion homomorphic images of A, then B ^ C if and only if d(B) = d(C) and B = C^ for almost all p. Let 5 and p THE CLASSIFICATION OF CERTAIN CLASSES OF TORSION FREE 649 C be torsion free. Then B ~ C is equivalent to the existence of a monomorphism ψ on B into C such that C/φ(B) is of bounded order. It is well-known that if B ϋ A and B = A, then A/B is a finite group. This has the important consequence that A £k B it and only if each group is imbeddable in the other one. B and C are quasi- equal, denoted by B = C, if there are positive integers n and m such that JB 2 nC Ξ2 mΰ. Two torsion free Z modules are quasi-equal if p they are quasi-equal as groups. The local-global relations give: A == B if and only if Z A = Z B for almost all p and Z A = Z B for all p. P P P P LEMMA 1. Let B and C be full subgroups of A. If B = C, then A/B έ A/C. Proof. Since Z B = Z^C for almost all p, (A/B) s Z A/Z B = P p P P Z A/Z C ^ (A/C),, for almost all p. Now C^mBSnC for some P P w, m > 0. Let J3' = mΰ. Since C/5' is bounded, the exact sequence id C/B'> >A/B'-»A/C shows that d(A/B') = d(A/C). Since A is tor- sion free, A/B ~ mA/mB = m{A/B') and so d(A/B) = d(A/Bf). Hence, s A/C. DEFINITION. k (A) = r(d(A/I)) where / is a full, free subgroup p p of A and s (A) = r{C[ pnA). p n Note that k(A) does not depend upon / by Lemma 1. As in [1] we p let δ (A) denote the maximal divisible subgroup of A(p), which is the p maximal divisible submodule of Aip) regarded as a Z{p) module. Thus, δ(A) is a Q{p) subspace of V(p). p LEMMA 2. (i) r(A) = r(A) + k(A) p p (ii) k (A) = r (δ(A)) p zW p (iii) s,(A) - r(δ(A) n F) p Proof. For (i), let / be a full, free subgroup of A such that (A/I) is divisible. Then (V/I)^ (A/I)@(V/A) and so r(F) = p p 9 P A:,(A) + r((F/A),). Since V/A-V/pA and T7pA[p]=-A/pA, r((V/A)) = ί 2 9 r{A/pA), which gives (i). To show (ii) it will be enough in view of (i) to show r(A) = r(A) +rw(δ(A)). Now Z{p) (g)A = d(Zp) ®A)(&F p z p where F is a free Z(2))-module [5, 44.2] and so r{F/pF) = rw{F). z Thus, it will be sufficient to show r^A) = r(F/pF). The exact sequence pA > > A -» A/pA implies 2r(p) ® pA > > Z(p) ® A -» (g) (A/pA) exact. Note that e(Zp) (x) pA) = p(Zp) ® A). Thus Z(p) (g) A/p(Z{p) ®A)~ Z{p) (g) (A/pA) = A/pA, which gives (ii). For (iii), note that Z(ΠpnA) = d(^A) = ί (A) Π 7". p n p COROLLARY 1. Let Bbe a full subgroup of A. Then the following 650 C. E. MURLEY conditions are equivalent: (i) (A/B) is reduced, (ii) d(A) = δ(B), p p p (iii) k(A) = k(B), and (iv) r{A) = r{B). p p p p LEMMA 3. Let B be a subgroup of A and H a characteristic such that {peΠ\H(p) = c«} = Π(A). Then the following are equivalent: (i) B = A[H] and τ*{A) ^ [H] (ii) A/B is torsion with (A/B) = ©V/1} Z(pH{p)) for p g Π(A) and p (A/B) = {0} for peΠ(A) p (iii) pH{p)ZA = Z B for H(p) < oo and Z A = ZBfor H(p) = oo. p P P p Proof. We give a cyclical proof. Assume (i). Then τ#(A) ^ [IT] gives A/B torsion. Since A[£Γ] = Γ\*πu)qHlq)A, Z B = Z(pH(p)A) q V p for p$Π(A) and ^β - Z A for peΠ(A). Thus, (A/5) = {0} for P p peΠ(A) and (A/5), = A/pH{p)A ~ ®^u) Z{pH{p)) for p£/7(A), which is (ii). Assume (ii). Then Z A = ^£ for peΠ(A). For pZΠ(A), P pHip)ZA S ^# S #*A and ZA/pΠ{p)ZA is a p-group, with the same p p p order as ^ A/Z £. Thus, ^{p)Z A = Z JB for p £ Π(A), which is (iii). As- P P p p sume (iii). Since A/A[H] is torsion, r*(A)^ [H]. Since 77(A) = 77(5), B= n (H { p )ZA V) Π 4H { )Z A A) nH { p )A A [ f l] which is (i). COROLLARY 2. 1/ τJA) ^ [fl], ίΛβ^ End (A) = End (A[7Ϊ]). Proo/. For ^eEnd(A), let φ' be the restriction of φ to A[f7]. Since A[ί7] is a full, fully invariant subgroup of A, φ —• 0' is a ring monomorphism into End (A[ί7]) For λ' e End (A[7Γ]), let λ be its unique extension to A into V. By Lemma 3, pH{p)ZA = Z A[H] 2 p P Z (λ(A[ί7])) = p^%λ(A) for p^77(A). Thus, Z,λ(A) Q Z A for all p P p and so λe End (A). 2* The quotient divisible core* We recall from [1] that A is a quotient divisible [QD] group if A has a full free subgroup I such that A/I is divisible. Note that A is a QD group if and only if for J a full free subgroup of A, A/J = ΰ φΓ where D is divisible and T is finite. The invariants introduced by Beaumont-Pierce in [1] to classify the QD groups in jy involve the following considerations. Let J^ (V) denote the lattice of all Q(p)-subspaces of V{p) and ^f(V) = P Tίp^f (V) the direct product of these lattices. If δej^(V), then the P p-component of 8 is denoted by δ and 3 is referred to as a QD in- p variant (associated with V). For φ a Q-automorphism of V, let φ{p) = φ®id ( ), which is a Q(p)-automorphism of Viv). Q P DEFINITION. Let δ, 5' e £f(V). Then δ ^ δ' if there is a Q-auto- morphism φ of Fsuch that Φ{p)(d) g δ' for all p. δ - δ; if δ ^ δ' and p p δ' ^ δ. For A full in V, let δ(A) 6 .Sf{V) such that δ(A) = δ(A). p p THE CLASSIFICATION OF CERTAIN CLASSES OF TORSION FREE 651 Let A and B be full QD subgroups of V. Then the Beaumont- Pierce QD Theorem [1, 5.25] states that: (i) A is imbeddable in B if and only if δ(A) ^ δ(B), (ii) A = B[A^B] if and only if δ(A) = δ(B)[δ(A) ~ δ(B)], (iii) For δe£f(V), there is a full QD subgroup A of V such that δ(A) = δ. DEFINITION. Let 7 be a full free subgroup of A and φ be the natural map A -» A/1. Then QD(A, I) = φ~ι{d{AII)). LEMMA 4. Let I and J be full free subgroups of A and let B be a full subgroup of A. Then: (i) B contains I and δ(A) = δ(B) if and only if QD(A, I) £ B (ii) if B is QD and δ(A) = δ(B), then QD(A, J) = B for some J (iii) QD(A, I) = QD(A, J) and A/QD(A, I) £ A/QD(A, J). Proof. First note that QD(A, I) is a full QD subgroup of A such that A/QD(A I) is reduced torsion. Part (i) is now immediate y from Corollary 1. For (ii), let J be a full free subgroup of B such that B/J is divisible. The first part gives QD(A, J) £ B and so B/J~ QD(A, J)/J0 B/QD(A, J), which shows that B/QD(A, J) is divisible. Since <?(£) - δ(QD(A, J)), B/QD(A, J) is reduced by Corollary 1. Hence, B — QD(A, J), which is (ii). For the first part of (iii), you may invoke the Beaumont-Pierce QD Theorem or more directly, note that QD(A I) + J is a QD subgroup of A which is quasi-equal to y QD(A, I). Since QD(A, I) + J 3 QD(A, I + J) 3 QZ)(Λ, I) by (i), QD(A, I) = QD(A, I + J). Thus, QD(A, I) = QD(A, J) by symmetry. The second part of (iii) is now immediate from Lemma 1. For the remainder of this section / will denote a full free sub- group of A. Note that A is a locally free group, i.e. Z A is a free V Z module for all p, if and only if QD(A, I) — I and A is a QD group p if and only if QD(A I) = A. The quasi-isomorphism class determined 9 by QD{A, I), which by Lemma 4 is independent of the choice of J, will be referred to as the QD core of A. The Beaumont-Pierce QD Theorem shows that two groups A and B have the same QD core if and only if δ(A)~δ(B). The quasi-isomorphism class determined by A/QD(A, J), which by Lemma 4 is independent of choice of /, is closely related to the Richman type of A. See [12] or [13]. Let Ajl = d(A/I) 0 T. Then T ~ A/QD(A, I) and r{T) ^ r,{A) v for all p by Lemma 2. Thus, for r(A) >0, (A/QD(A, I)) = ®ϊg[A) Z(pa^v)) p p where 0 ^ ^(p) ^ α(j)) < °o for i > i and (A/QZ)(^4, J)) = {0} for y p r,(A) = 0. 652 C. E. MURLEY DEFINITION. For peΠ, let H*{A, I)(p) = a{p) if r(A) > 0 and λ p oo otherwise, £Γ*(A,1)(p) = α,(p) where s = r(A) if r (A) > 0 and p p oo otherwise, and τ*(A) = [H*(A, I)]. Lemma 4 shows that the types τ*(A) and [H*(A, I)] are inde- pendent of the choice of I. The identification of [if* (A, I)] in (i) of the following was also noted by Warfield [13, p. 194]. LEMMA 5. (i) τ*(A) = [H*(A, I)] (ii) r*(A) = τ^{B) and τ*(A) = τ*(B) whenever A±B (iii) A is a QD group if and only if τ*(A) is non-nil. Proof. For (i), let Heτ^A) and J be a full free subgroup of A[H\. To see that H*(A, I) ^ H, refer to Lemmas 3 and 4 to note that QD(A, I) S A[H] and consider the orders of the finite p-groups (A/QD(A, I)), (A/A[H]) and the natural map A/QD(A, I)-» A/A[H]. p P On the other hand, let B be a group such that QD(A, I) g B g A and for p $ /7(A), (A/B) = φrp{A) Z(pa) where a = H*(A, I)(p). Then p ^"•(-4.) ^ [H*(A, I)] by Lemma 3, which gives (i). Finally, (ii) is an easy computation while (iii) is immediate from the definitions. We have shown that every full subgroup A oί V is an extension of a QD group B by a reduced torsion group C and that B and C are unique up to quasi-isomorphism. On the other hand, let B be a full QD group in V and C be a reduced torsion group such that r(B) ^> p r(C) for all p. Then there is a full subgroup A in V which is an p extension of B by C. This may be seen by observing that r(B) = p r((V/B)) for all p and letting A be the inverse image in V of an p appropriate subgroup of the divisible group V/B. Now suppose both A and A! are extensions of B by C. Then it is easily seen that A and A' have the same QD core, τ*(A) = τ*(A') and τ (A) = r (A'). # # Thus, A = A' whenever r(2?) = 1. In the next section we study a class of groups A which are determined up to quasi-isomorphism by B and C In contrast we give the following example of two non- quasi-isomorphic groups A and A! with the same QD core and A/QD(A, I) s A'IQD(A\ J). EXAMPLE 1. Let A and A! be locally free, completely decom- posable groups of rank 2 whose type sets, denoted by T{—), satisfy T(A) Φ T(A'), sup T(A) - sup Γ(A'), and r^A) = inf Γ(A) = inf T(A') = τ^A'). Such pairs of groups exist in abundance. Let I and J be full free subgroups of A and A' respectively. Then QD(A, I) = I and ', J) — J (since the groups are locally free). Thus, QD(A, /) = /, J). A simple computation shows r*(A) = sup T(A) = τ*(A'). THE CLASSIFICATION OF CERTAIN CLASSES OF TORSION FREE 653 Hence, A/QD(A, I) ς± A'/QD(A\ J) (since A and A' have rank 2) and A is not ± to A' (since T{A) Φ T(A')). 3- The Class &. DEFINITION, & = {A e jzf | r*(A) = Note that τ*(A) = τ*{A) describes the condition that for I a full free subgroup of A, (A/QD(A, I)) is a direct sum of r(A) isomor- p p phic cyclics for almost all p. The finite rank QD groups and the finite rank, homogeneous, completely decomposable groups are examples of groups in &. In fact, the locally free groups in & are necessarily homogeneous, completely decomposable groups [13, Corollary 5]. Since r*(—) and τ^(-) are quasi-isomorphism invariants [Lemma 5], Ae& whenever A = B and B e &. Finally, we mention that & is closed with respect to direct summands and finite direct sums of the form ®? Ai where A^ & and τ*(Ai) = τ*(A) for all i, j. =1 3 LEMMA 6. Let He τ*(A). Then 4 e^ if and only if QD(A, J) = A[H) for some full free subgroup J in A. Proof. Let A G^ and let J be a full free in A[H]. Then QD{A, I) <ΞΞ A[H] by Lemmas 3 and 4. Since H ~ H*(A, I) by Lemma 5, Lemma 3 (ii) gives (A/A[H]) = (A/QD(A, I)) for almost all p. This P p says that A[H]/QD(A, I) is finite. Thus, A[H] is a QD group such that A/A[H] is reduced torsion. It follows from Lemma 4 that A[H] = QD(A, J) for some full free J in A. The converse is immediate from Lemma 3. DEFINITION. A type τ and a QD invariant 8 e £f(V) are compati- ble if the pth-component of r is oo if and only if δ = V{v). Note that p τ*(A) and 8(A) are compatible for A THEOREM 1. Let A and B be groups in & which are full in V. (i) There is an imbedding of A in B if and only if 8(A) ^ δ(B) and τ*(A) ^ τ*(B). (ii) A = Bif and only if 8(A) = δ(B) and τ (A) = τ (B). % % (iii) A± B if and only if 8 (A) - δ(B) and τ*{A) = τ*(B). (iv) J/ δejzf(V) and τ is a type compatible with δ, ί/ιe% ίfcere is α group C in & which is full in Vsuch that 8(C) = 8 and τ*(C) = r. Proo/ o/ ( i ). Let φ be an imbedding of A in ΰ and assume that φ is extended to a Q-automorphism of V. Then it is clear that 654 C. E. MURLEY Φ{p)(δ(A)) S δ(B) for all p, i.e. δ(A) ^ δ(B). Let I be a full free sub- p p group of A. Then ψ(I) is a full free subgroup of B and A/I = Φ(A)/φ(I) s B/φ(I). A modest computation shows II*(0(A), 0(1)) ^ #*(£, 0(1)), i.e. τ,(A) ^ r*(B). Conversely, let IZieτ*(A) and H €τ*(B) such that i^ ^ i?. Use 2 2 Lemma 6 to obtain full free subgroups I and J of 4 and 5 respec- tively such that A[H] = QD(A, I) and B[H] = QD(B, J). Since λ 2 δ(B), the Beaumont-Pierce QD Theorem gives an imbedding φ: B[H]. Assume that φ is uniquely extended to A into V and so for 2 p e /7(JB), Z0(A) S 7= ZB. For p $ Π(B), pH^p)Zφ{A) = Zφ(A\Έ$) £ P P p p i^J3 [iJ] — pH2{p)ZB by Lemma 3. Since division is unique in 2 p Tζ Zφ(A) £ p%£ £ Z£ where ί = iJ(p) - H^p). Hence, 0(A) £ β. p p 2 Proof of (ii). The "only if" part is immediate. For the con- verse let H = H and I, J be as in the proof of (i). Since δ(A) = x 2 δ(JS), A[£ΓJ = QD(A, /) = QD(B, J) = B[H] and so the imbedding φ in 2 the previous part may be chosen to be a left multiplication by some positive integer n. The argument in the previous part now gives nA £ B. By symmetry, mB £ A for some m > 0 and so A = B. Proof of (Hi). Since A ^ J5 if and only if each group is imbedd- able in the other, (iii) is immediate from (i). Proof of (iv). Let B be a full QD subgroup of V with δ(B) = δ. Let iJe τ and C be a subgroup of V containing B such that (C/B) = p 0V5) Z(pH{p)) for p0Π(B). Note that such a C exists since F/£ is divisible with r((V/B)) = r(5). Since τ and δ are compatible, C/B p p is reduced torsion and Lemma 4(ii) gives B = QD(C, I) where / is a full free subgroup of B such that B/I is divisible. Thus, Ce & with δ(C) = δ and r (C) - τ. # REMARK 1. Let A be a QD group and H the characteristic such that H{p) = oo if d(A) = Vip) and H(p) = 0 otherwise. Then by p Lemma 5 (iii) [H] = ^(A). Thus, r^(A) may be recaptured from δ(A) whenever A is a QD group. This shows that Theorem 1 is a gener- alization of [1, 5.25]. We mention that the Warfield Duality [13] may be used to show that for D a torsion free, rank 1 group with τ*(D) = τ+(A) and I a full free subgroup of A, then the following are equivalent: (i) Ae^, (ii) QD(A, I) ^ Horn (D, A), (iii) A έ ΰ® QD(A, I). This, of course, yields another proof of Theorem 1. LEMMA 7. If I is a full free subgroup of A and A e &, then Q (x) End (A) ~ Q (x) End (QD(A, I)). THE CLASSIFICATION OF CERTAIN CLASSES OF TORSION FREE 655 Proof. Let Heτ*(A) and J be a full free subgroup of A such that A[H] = QD(A, J) [Lemma 6]. Thus, A[H] = QD(A, I) and so Q <g) End (4)sQ(g) End (A[H]) = Q (g> End (QD(A, /)) by Corollary 2. For the following corollary, recall that A is quasi-decomposable if it is quasi-equal to the direct sum of two nonzero torsion free groups. A is strongly indecomposable if it is not quasi-decomposable. COROLLARY 3. Let I be a full free subgroup of A and A e &. Then the following are equivalent: (i) A is quasi-decomposable (ii) QD(A, I) is quasi-decomposable (ίii) there are nonzero subspaces U and W such that V = U 0 W and δ (A) = δ(A) Π U(p) 0 δ(A) Π W{p) for all p. p p p Proof. Since a group B in Jzf is quasi-decomposable if and only if Q (x) End (B) is decomposable as a module over itself [11], the equiva- lence of (i) and (ii) is immediate from Lemma 7. (iii) is a necessary and sufficient condition for a QD group in jy to be quasi-decom- posable [1, 5.26]. Since δ(A) = d(QD(A /)), (ii) and (iii) are equiva- y lent. Note that the quotient-divisibility of A must be added to the hypotheses of [1, 5.26]. 4. The Class if- DEFINITION. & = {Aej^\r(A) ^ 1 for all peΠ}. p We note that if is a subclass of & which contains the torsion free, rank one groups. Furthermore, the reduced groups A in g7 are up to isomorphism precisely the finite rank, pure subgroups of T[ eπZ(p\ the Z-adic completion of the integers. The class ί? is closed P with respect to pure subgroups, torsion free homomorphic images and tensor products. Here we make use of the facts that for 4,.ΰ6 J/, r(A (x) B) = r^r^B) and for B pure in A, r (A) = r (B) + r(A/B). p p p p Recall that a group A in Jzf is cohesive iί Ae& and s (A) = 0 for p p ί i7(A) [4]. If A is a non-cohesive group in g% i.e. 0 < s (A) < r(A) p for some p, then A is not homogeneous (since there are 0 ^ x, ye A such that HA(x)(p) = <>o and HA(y)(p) < <χ>). Thus, the homogeneous groups in έf are cohesive. On the other hand, Theorem 4 in [4] shows the existence of homogeneous and non-homogeneous cohesive groups of any rank greater than one. Richman's special groups [12] are a subclass of the homogeneous groups in g7. Reduced groups in £? can be decomposable. For example, if {Π\ 77"} is a nontrivial partition of 77, Z(Π') = Π eπ Z , Z(Π") = ΠPSΠ Z , and A - Z{Π') 0 Z(Π"), then P p P Aeg7. On the other hand, the reduced cohesive groups in i? are