Strongly and solidly ?1-weak p^?·2+n-projective abelian p-groups.pdf 1. Introduction and terminology Let all groups into consideration be p-primary abelian, where p is a fixed prime integer, written additively as it is customary. As usual, for some ordinal a > 0 and a group G, we state the a-th Ulm subgroup paG, consisting of all elements of G with height > a, inductively as follows: p0G = G, pG = {pg | g e G}, paG = p(pa-1G) if a - 1 exists (so a is non-limit) andpaG = Op 0 recall that a group G is pm+n-projective if G/P is Y-cyclic for some pn-bounded P < G. Note the crucial facts from [1] that if G is either Y-cyclic or pm+n-projective, then so is G/F for every finite F < G. We continue with two crucial concepts investigated in [3] in details. • A group G is said to be weakly pm'2+n-projective if there is a pm+n-projective subgroup H < G such that G/H is Y-cyclic. It was established in [3] that these groups are pm2+n-bounded. • A group G is said to be o>1-weakly pm'2+n-projective if there is a countable subgroup K < G such that G/K is weakly pm 2+n-projective. It was proved in [3] that for such a group G its subgrouppm'2+nG is always countable. However, this class of groups is quite large, and it will be better to consider some its restricted modifications by exploiting in various aspects the ”niceness” property. Recall that a subgroup N of a group G is nice if, for each limit ordinal т , the equality fla1-pm-projective. Definition 1.2. A group G is said to be solidly o1-weak pro'2+n-projective if it contains a countable nice subgroup M such that G/M is weakly pm2+n-projective. Definition 1.3. A group G is said to be nicely o1-weak pro'2+n-projective if it contains a weakly pm'2+n-projective nice subgroup Q such that G/Q is countable. The goal of the present paper is to give a comprehensive study of these three concepts, thus somewhat enlarging the results from [2], [3], and [4]. The work is organized as follows: in the next two sections, we state some elementary and useful properties of the new group classes. After that, we establish our basic results. In the final section, we list some interesting left-open questions. And so, we come to our first working section. 2. Elementary properties Here we shall quote some elementary but helpful properties like these: (1) Strongly oh-weak pm2+n-projective groups are Oh-weakly pm2+n-projective. (2) Solidly Oh-weak po2+n-projective groups are o1-weakly po2+n-projective. (3) Nicely o1-weak po2+n-projective groups are o1-weakly po2+n-projective (this follows from Theorem 2.2 (e) of [3]). (4) Weakly po2+n-projective groups are both strongly o1-weak po2+n-projective and solidly Oh-weak po2+n-projective. (5) Strongly o1-pO+n-projective groups are strongly o1-weak po2+n-projective. (6) Nicely o1-pO+n-projective groups are nicely o1-weak po2+n-projective. (7) If pmG = {0}, then G is strongly o1-weak po2+n-projective G is solidly ю1weak po 2+n-projective G is nicely o1-weak po 2+n-projective G is weakly po 2+nprojective. In fact, in [3] was showed even that pO-bounded o1-weak po2+n-projective groups are weakly po2+n-projective. The following relationship sounds interesting. Proposition 2.1. If G is a strongly o1-weak pao2+n-projective group, then G is nicely o1-weak pm'2+n-projective. Proof. Write G/N = (K/N) Ѳ (S/N) for some pO+n-projective nice subgroup N of G with N < K and N < S, where the first term of the direct decomposition is countable whereas the second one is E-cyclic. Thus G/S = (G/N)/(S/N) = K/N is countable. But S/N is nice in G/N as a direct summand, whence by virtue of [7] we derive that S is nice in G. Moreover, S is by definition weakly po2+n- projective, as required. ■ 3. Some useful preliminaries The following three affirmations, dealing with weakly po2+n-projective groups, seem not to appear in [3], and so we will document them here. Proposition 3.1. (i) The group G is weakly pm'2-projective if and only if pnG is weakly pm'2-projective for some n e N. (ii) If G is weakly pm'2-projective and T < G with pnT = {0}, then G/T is weakly pao2+n-projective. Strongly and solidly юі-weak p m2+n-projective abelian p-groups 7 Proof. (i) (^) Suppose there is a E-cyclic subgroupX of G such that G/Xis E-cyclic. Hence pn(G/X) = (pnG + X)/X = pnG/(pnG П X) and pnC П X are both E-cyclic groups being subgroups of G/X and X, respectively (see, e.g., [7]). (^) Suppose that there exists a E-cyclic subgroup Y of pnG, and hence of G, with pnG/Y=pn(G/Y) also a E-cyclic group. But consulting with [7], the quotient pn(G/Y) being E-cyclic implies the same for G/Y as well. This gives the wanted result. (ii) Let G/U be E-cyclic for some E-cyclic subgroup U . Thus U + T is also E-cyclic (see [7]) and U ' = (U + T)/T = U/(U П T) is thereforepm+n-projective. But (G/T)/((U + T)/T) = G/(U + T) = (G/U)/((U + T)/U) is p“+n-projective, because (U + T)/U is pn-bounded. Denote G/T = G'. Since G'/U' is pm+n-projective, there is Z’ < G' with U' c Z' and pnZ' c U' such that (G'/U ')/(Z'/U ') = = G'/Z' is E-cyclic. But pnZ' is pm+n-projective, whence so is Z'. Finally, G' = G/T is weakly pm 2+n-projective, as claimed. ■ Theorem 3.2. The following four points are equivalent: (a) G is weakly pm'2+n-projective; (b) there exists a pm+n-projective subgroup P < G such that G/P is E-cyclic; (c) there exists a pn-bounded subgroup T < G such that G/T is weakly pm'2-projective; (d) there exist a pn-bounded subgroup L and a weakly pm'2-projective group S such that G = S/L. Proof. (a) (b) is just the definition. (b) ^ (c). Assume P/X is E-cyclic for some pnX = {0}. Thus G/P = (G/X)/(P/X) is E-cyclic, whence G/X is by definition weakly pm2-projective, as expected. (c) ^ (b). Let A/T be E-cyclic for some A < G containing T such that (G/T)/(A/T) = G/A is also E-cyclic. But it is plainly seen that A is pm+n-projective, as required. The implication (d) ^ (a), or its equivalence (d) ^ (b), follows from Proposition 3.1 (ii). So, we consider the reverse implication (a) ^ (d) or its tantamount relationship (c) ^ (d). To that aim, if X is a group with pnX = G, then let S = X/T. Consequently, pnS = pnX/T = G/T is weakly pm 2-projective by hypothesis. Referring to Proposition 3.1 (i), the last condition forces that S is weakly pm2-projective. Letting L = X[pn]/T c S[pn], we deduce that S/L = X/X[pn] = pnX = G, proving the desired relation. ■ Lemma 3.3. If A is a weakly pm'2+n-projective group and F < A is finite, then A/F is also weakly pm'2+n-projective. Proof. Write A/B is E-cyclic for some pm+n-projective subgroup B. Since (F + B)/B = = F/(FnB) is obviously finite, one sees that (A/B)/(F+B)/B = A/(F+B) = (A/F)/(F+B)/F is E-cyclic. However, (F + B)/F = B/(B П F) is pm+n-projective too (see [1]), as required. ■ 4. Main Results The following two assertions strengthen point (5) listed above. Proposition 4.1. If G is a strongly щ-weakpm'2+n-projective group andpm+nG = {0}, then G is weakly pm'2+n-projective. Proof. Write G/N is the direct sum of a countable group and a E-cyclic group for some nice pm+n-projective subgroup N of G. Since pm+n(G/N) = (pm+nG + N)/N = {0}, we have that G/N is pm+n-projective. Hence there is a subgroup X < G containing N such that pnX c N and (G/N)/(X/N) = G/X is E-cyclic. Since pnX is pm+n-projective, we infer that so is X, as required. ■ Peter V. Danchev 8 Proposition 4.2. If G is a strongly щ-weak pm'2+n-projective group and pmG is finite, then G is weakly pm'2+n-projective. Proof. Let G/N be the direct sum of a countable group and a E-cyclic group for some pm+n-projective nice subgroup N of G. Therefore, (G/N)/pm(G/N) = (G/N)/(pmG+N)/N = = G/(pmG+N) is E-cyclic. Denote V=pmG+N, and hence V/T = (pmG + N)/T = = [(pmG + T)/T ] + [N/T ], where T < N with the property that pnT = {0} and N/T is E-cyclic. Since (pmG + T)/T = pmG/(pmG П T) is finite, it follows that V/T is E-cyclic. Thus V is pm+n-projective and G/V is E-cyclic, as required. ■ We continue with Proposition 4.3. If G is a strongly щ-weak pm'2+n-projective group and pmG is countable, then G/pmG is weakly pm'2+n-projective. Proof. Observe that (N +pmG)/pmG = N/(N np“G) = [N/pmN ]/[(N np“G)/p“N ] is separable being embedded in G/pmG, and thus it is pm+n-projective according to Theorem 4.2 of [5]. But pm(G/N) is countable being contained in a direct summand of G/N, whence it easily follows that (G/N)/pm(G/N) = (G/N)/((pmG + N)/N) = G/(pmG + N) = = (G/pmG)/((pmG + N)/paG) is E-cyclic, as required for the factor-group G/pmG to be weakly pm 2+n-projective. ■ Remark 1. The last statement follows also from Theorem 2.4 in [3], but the stated above argument gives a new more simple and conceptual proof. Analyzing the corresponding definitions, especially Definition 1.2, and again utilizing the same theorem, we then can say even a little more: Theorem 4.4. If G is a group such that pmG is countable, then the following points are equivalent: (i) G is щ-weaklypm'2+n-projective; (ii) G is solidly щ-weakpm'2+n-projective; (iii) G/pmG is weakly pm'2+n-projective. Thus Proposition 4.3 can be extended to nicely ra1-weak pm2+”-projective groups (compare with Proposition 2.1 quoted above). However, when the subgroups paG are finite for some infinite ordinal a, we obtain the following strengthening. Proposition 4.5. If G is a strongly щ-weak pm'2+n-projective group and paG is finite for some a > Ю, then G/paG is strongly щ-weakpm'2+n-projective. Proof. Given G/N = (C/N)®(S/N), where C/N is countable and S/N is E-cyclic for some pm+”-projective subgroup N of G which is nice in G. But pa(G/N) = pa(C/N) and thus (G/N)/pa(G/N) = [(C/N)/pa(C/N)] Ѳ (S/N) and therefore, since pa(G/N) = (paG + N)/N, we obtain that G/(paG + N) = (G/paG)/(paG + N)/paG is the direct sum of a countable group and a E-cyclic group. However, (paG + N)/paG is nice in G/paG because paG+N is so in G (see, e.g., [7]), and moreover (paG + N)/paG = = N/(N П paG) ispm+”-projective since N П paG is finite (see, for instance, [1]). ■ Proposition 4.6. If G is a solidly щ-weak pm'2+n-projective group and paG is finite for some a > Ю, then G/paG is solidly щ-weakpm'2+n-projective. Proof. Let M be a countable nice subgroup of G such that G/M is weakly pm'2+n-projective. Since pa(G/M) = (paG+M)/M = paG/(paGnM) is finite, according to Lemma 3.3, we deduce that Strongly and solidly юі-weak p m2+n-projective abelian p-groups 9 (GIM)lpa(GIM) = GI(paG + M) = (GlpaG)l(paG + M)/paG is weakly pm'2+”-projective as well. Moreover, (paG + M)IpaG = M/(M П paG) is countable and nice in GIpaG (cf. [7]), as desired. ■ Proposition 4.7. If G is a nicely оу-weak pm'2+n-projective group and paG is finite for some a > ю, then GIpaG is nicely щ-weakpm'2+n-projective. Proof. Write GIQ is countable for some nice weakly pm'2+n-projective subgroup Q. Likewise, in virtue of Lemma 3.3, the quotient (Q + paG)Ip“G = QI(Q П paG) is again weakly pm'2+n-projective. We also derive that GI(Q + paG) = (GIpaG)I(Q + paG)IpaG is countable. But (Q + paG)IpaG is nice in GIpaG by [7], as wanted. ■ The following technicality is well-known, but we list and prove it here only for the sake of completeness and for the convenience of the reader. Lemma 4.8. If A is a L-cyclic group and C < A is its countable subgroup, then AIC is a direct sum of a countable group and a L-cyclic group. In particular, if C is nice in A, then AIC is also a L-cyclic group. Proof. Since C is countable, there exists a countable subgroup K of A with the property that K з C and A = K 0 T for some T < A. Therefore, AIC = (KIC) 0 T , where C is nice in K. Thus, KIC is a separable countable group and hence a L-cyclic group, as wanted. ■ For arbitrary infinite ordinals a and countable Ulm subgroups paG, we have the following: Proposition 4.9. If a > ю and G is a strongly щ-weak pm'2-projective group such thatpaG is countable, then GIpaG modulo a countable subgroup is nicely ol-weakpm'2-projective. Proof. Suppose that there is a nice L-cyclic subgroup N of G such that GIN is the direct sum of a countable group and a L-cyclic group. Since (paG + N)IN=pa(GIN) is countable and contained in the countable direct summand of GIN , it easily follows that GI(paG + N) = (GIN)Ipa(GIN) = (GIpaG)I(paG + N)IpaG is again the direct sum of a countable group and a L-cyclic group. Moreover, (paG + N)IpaG is nice in GIpaG (see [7]) and, because N П paG is countable, one can infer by Lemma 4.8 that (paG + N)IpaG = NI(N П paG) is the direct sum of a countable group and a L-cyclic group, say (paG + N)IpaG = K 0 S, where the first term K is countable and the second one S is L-cyclic. Consequently, denoting Ga = GIpaG, we write GaI(K 0 S) = [RI(K 0 S)] 0 [VI(K 0 S)], where RI(K 0 S) is countable while VI(K 0 S) is L-cyclic. Since K 0 S is nice in Ga, it follows from [7] that V is also nice in Ga. Besides, GaIV is countable. On the other hand, both (VIK)I(K 0 S)IK = VI(K 0 S) and (K 0 S)IK = S must be L-cyclic, whence VIK must be weakly pm2-projective. So V is ral-weakly pm 2-projective. Finally, one sees that (GaIK)I(VIK) = GaIV, as required. ■ We will now consider how the three new properties are inherited by the action on Ulm subgroups. Proposition 4.10. If G is strongly ol-weak pm'2+n-projective, then so is paG for any ordinal a. Proof. Let there exist a nice pm+”-projective subgroup N of G such that GIN is the direct sum of a countable group and a L-cyclic group. Therefore, N П paG is by [7] nice in paG and is also pm+”-projective being a subgroup of N. Moreover, paGI(paG П N) = = (paG + N)IN c GIN is ю-totally L-cyclic as well (cf. [6]), thus proving the assertion, as promised. ■ Peter V. Danchev 10 Proposition 4.11. If G is solidly щ-weak pm'2+n-projective, then so is paG for any ordinal a. Proof. There is a countable nice subgroup M of G such that G/M is weakly pm'2+n-projective. Thus, in view of [3], we have thatpa(G/M) = (paG +M)/M = paG/(paG П M) is also weakly pm 2+n-projective. But M П paG is countable and nice in paG (cf. [7]), as required. ■ Proposition 4.12. If G is nicely щ-weak pm'2+n-projective, then so is paG for any ordinal a. Proof. Write G/Q is countable for some nice pm 2+n-projective subgroup Q of G. Therefore, pa(G/Q) = (paG + Q)/Q = paG/(paG П Q) is countable, where paG П Q is nice inpaG (see [7]) andpaG П Q ispm 2+n-projective being a subgroup of Q (see [3]). ■ The next technicality is well-known but we, however, will give a proof for the sake of completeness and for the readers’ convenience. Lemma 4.13. If C is a countable group and L is a L-cyclic group, then there are a countable group K and a L-cyclic group S such that C + L = K ® S. Proof. Since C П L c L' for some countable subgroup L' of L with L = L' Ѳ L”, we have that C + L = C + L' + L” = (C + Lr) Ѳ L". In fact, x e (C + L') П L” gives that x = c + b, where b e L' and c e C. Thus x - b = c e C П L and hence x - b e L' which forces that x e L" П L' = 0, as required. Putting now C + L' = Kand L" = S, we are set. ■ The following two technical claims are pivotal (see also [4]). Lemma 4.14. Suppose that a is an ordinal, and that G and F are groups where F is finite. Then the following formula is fulfilled'. pa(G + F) = paG + F П pa(G + F). Proof. We will use a transfinite induction on a. First, if a - 1 exists, we have pa(G + F) = p(pa-1(G + F)) = p(pa-1G + F П pa-1(G + F)) = p(pa-1G) + p(F П pa-1(G + F)) c paG + F П p(pa-1(G + F)) = paG + F П pa(G + F). Since the reverse inclusion ”з” is obvious, we obtain the desired equality. If now a-1 does not exist, we have that pa(G+F) = Пр^ (p e(G+F)) c c Пp

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