Realization theorems for valuated <span class="mathjax-formula">$p^n$</span> -socles

Realization Theorems for Valuated p

-Socles

PATRICKW. KEEF

ABSTRACT- Ifnisa positive integer andpisa prime, then a valuatedpⁿ-socle is said to ben-summable if it is isometric to a valuated direct sum of countable valuated groups. The functions fromv1 to the cardinalsthat can appear asthe Ulm function of ann-summable valuatedpⁿ-socle are characterized, as are then- summable valuatedpⁿ-socles that can appear as thepⁿ-socle of some primary abelian group. The second statement generalizes a classical result of Honda from [9]. Assuming a particular consequence of the generalized continuum hypoth- esis, a complete description is given of then-summable groups that are uniquely determined by their Ulm functions.

0. Terminology and introduction

Except where specifically noted, the term ``group'' will mean an abelianp- group, wherepisa prime fixed for the duration of the paper. Our terminology and notation will be based upon [6]. A group isS-cyclicif it isisomorphic to a direct sum of cyclic groups. We also make use of concepts related tovaluated groupsandvaluated vector spacesthat can be found, for example, in [16] and [7], and that we briefly review: Let O be the classof ordinalsand O1 ˆ O [ f1g, where we agree thata51for alla2 O1. Avaluationon a groupV isa functionj j_V :V! O₁such that for everyx;y2V,jxyj_V minfjxj_V;jyj_Vg and jpxj_V>jxj_V. As a result, for all a2 O₁, V(a)ˆ fx2V :jxj_V agisa subgroup ofVwithpV(a)V(a‡1). We sayV isa- boundedifV(a)ˆ f0g; thelengthofVisthe leastasuch thatV(a)ˆV(1).

A homomorphism between two valuated groups isvaluatedif it doesnot decrease values and an isometryif it isbijective and preservesvalues. If fV_ig_i2I, is a collection of valuated groups, then the usual direct sum, V ˆL

i2IV_i, hasa natural valuation, whereV(a)ˆL

i2IV_i(a) for everya2 O₁. (*) Indirizzo dell'A.: Department of Mathematics, Whitman College, Walla Walla, WA 99362, USA.

E-mail: [email protected]

IfW isany subgroup of V, then restrictingj j_V toW turnsW into a valuated group withW(a)ˆW\V(a) for alla2 O1. A valuated groupW with pW ˆ f0g iscalled a valuated vector space; so eachW(a) will be a subspace ofW. We say a valuated vector space isfreeif it isisometric to a valuated direct sum of cyclic groups (of orderp). IfV isa valuated group, then itssocleV[p]ˆ fx2V:pxˆ0gisa valuated vector space, andVis summableifV[p] isfree. A groupGisa valuated group using the height function (also denoted byj j_G) asitsvaluation; in thiscaseG(a)ˆp^aG, and Gissaid to beseparableif it isv-bounded. Ifnisa fixed positive integer, it followsthat thepⁿ-socleof G, writtenG[pⁿ]ˆ fx2G:pⁿxˆ0g;can be viewed asa valuated group.

In [4], an1-bounded valuated groupVwasdefined to be avaluated pⁿ- socle if pⁿV ˆ f0g and for every x2V[p^{n 1}] and every ordinalb5jxj_V, there isa y2V with xˆpy and b jyj_V. It easily follows that an 1- bounded valuated vector space is a valuatedp-socle. Thepⁿ-socle of a re- duced groupGisalwaysa valuatedpⁿ-socle, and ifV ˆG[pⁿ], then we will sayGissupportedbyV. A valuatedpⁿ-socle isrealizableif it issupported by some reduced group. [The parallel requirements thatV be1-bounded and thatGbe reduced are convenient, but not strictly speaking necessary.]

LetCbe the class of all cardinals. IfVisa valuated group, then thea-th Ulm invariantofV is

f_V(a)ˆr(V(a)[p]=V(a‡1)[p])2 C:

We call f_V :O ! CtheUlm functionofV. Another definition of the Ulm function of a general valuated group isgiven in [10], and it iseasy to verify that these agree for valuatedpⁿ-socles. In general, when we say f :O ! C isa function, we mean that itssupport iscontained in some ordinald, and we identify f with itsrestriction f :d! C.

A valuated pⁿ-socle is said to be n-summableif it isisometric to the valuated direct sum of a collection of countable valuated groups (each of which will also be a valuatedpⁿ-socle). It was shown in [4] that the theory of n-summable valuated pⁿ-socles parallels the theory of dsc groups (i.e., direct sums of countable groups - see Chapter XII of [6] for standard re- sults on these groups). For example, the following is ([4], Theorem 2.7), which parallels([6], Theorem 78.4).

THEOREM0.1. Suppose V and W are n-summable valuated pⁿ-socles.

Then there is an isometry V W iff their Ulm functions agree,i.e., f_V ˆf_W.

The parallel between n-summable valuated pⁿ-socles and dsc groups can be extended. A subgroupXof a valuated groupVisniceif every coset a‡Xhasan element of maximal value. Anice composition seriesforVis an ascending chain of nice subgroupsfX_i:idgsuch that

(C1) X0ˆ f0g,X_dˆV;

(C2) for alli5d,X_i‡1=X_iZ_p;

(C3) for all limit ordinalsld,X_lˆ S

i5lX_i.

Anice systemforVisa collectionN of nice subgroups ofV such that (S1) f0g 2 N;

(S2) N is closed under group sums;

(S3) IfSV iscountable, thenSNfor some countableN2 N. If V isa valuated pⁿ-socle, let l_V be the unique ordinal such that jVj_V ˆ fjxj_V :x2Vg O₁ isorder-isomorphic tol_V[ f1g. The follow- ing is([4], Theorem 2.1), which parallels([6], Theorem 81.9).

THEOREM0.2. Suppose V is a valuated pⁿ-socle andlV v1. Then the following are equivalent:

(a) V is n-summable;

(b) V has a nice system;

(c) V has a nice composition series.

If l_V v1 and f:jVj_V !l_V[ f1g isan order-preserving bijection andjxj⁰_V ˆf(jxj_V) for everyx2V, thenVisalso a valuatedpⁿ-socle using j j⁰_Vand virtually anything that istrue using one valuation (e.g., thatVisn- summable) is also true using the other. It therefore makes sense to restrict our attention to thosen-summable valuatedpⁿ-socles that arev1-bounded.

This leads to two of the main questions that are addressed in this paper:

(1) If f :v1! C isa function, when doesf ˆfV for some n-summable valuatedpⁿ-socleV?

(2) IfV isanv1-boundedn-summable valuatedpⁿ-socle, when isVrealiz- able?

Complete answers are given to both questions. For obvious reasons, a function satisfying (1) will be calledn-summable. Section 1 is a discussion of n-summable functions, which are characterized in Theorem 1.10. This combinatorial condition, which we describe later, can be viewed as a gen- eralization of the classical notion of an admissiblefunction (see [6], The- orem 83.6).

Section 2 is a consideration of question (2). In Theorem 2.11 it is shown thatVisrealizable iff for every countable limit ordinalland every a5lwe have

X

l‡n 1b5l‡v

f_V(b) X

a5b5l

f_V(b)_@0 :

Naturally, a groupGisn-summable ifG[pⁿ] is n-summable as a val- uatedpⁿ-socle. These groups are considered in [5], [13], [14] and [15]. Ann- summable group will always be summable (since a countable valuated vector space is free), and so p^v1Gˆ f0g (see, for example, [6], Theo- rem 84.3). Therefore, anyn-summable valuatedpⁿ-socle that is realizable must be v₁-bounded; this is another reason for restricting to the v₁- bounded case. It is known that a given valuated vector space is often supported by different groups that are not isomorphic. This suggests a third question.

(3) IfVisann-summable valuatedpⁿ-socle, when isVuniquely realizable, in the sense that any two groups supported byV are isomorphic?

Since n-summable valuated pⁿ-socles are classified by their Ulm functions, the last question can be restated as follows:

(3⁰) Describe then-summable groupsGthat are uniquely determined by their Ulm functions; that is, those that have the property that ifG⁰isan- othern-summable group with fGˆfG⁰, thenGisisomorphic toG⁰.

Assuming a natural statement regarding cardinal arithmetic that is a consequence of the generalized continuum hypothesis, this question is answered in Theorem 3.4. With this cardinality assumption, it is shown thatV isuniquely realizable iffV(v‡n 1) iscountable; and in thiscase, every groupG supported byV will be a dsc group. In particular, when nˆ1, Corollary 3.5 states that these groups agree with those described in ([2], Theorem 2.6).

Theorem 2.11 (i.e., the solution to the above question (2)) is a gen- eralization of the classical ``Existence Theorem for Principal p-Groups'' from [9]; in fact, for nˆ1, it reduces to precisely this result. However, there are several important differences. First, it is fairly clear that ifnˆ1, thenanyfunction f :v1! Cwill satisfy (1); i.e., any such function is the Ulm function of some free valuated vector space. On the other hand, if n>1, then f will be the Ulm function of ann-summable valuatedpⁿ-socle iff it satisfies Theorem 1.10. Second, the proofs of our main results are

considerably less complicated; i.e., a few pages in length, as opposed to the over twenty pagesneeded to prove the Existence Theorem in [9]. And third, we can apply our techniquesto answer question (3) whenever our condition on cardinal arithmetic holds; the latter question was not even considered in [9].

1. Realizing Ulm Functions

In this section we give an explicit description of those functions fromv₁ to the cardinalsthat aren-summable, in that they can appear as the Ulm function of an n-summable valuated pⁿ-socle. By considering (valuated) direct sums, it follows that the sum of a collection ofn-summable functions isn-summable.

Ifaisan ordinal, thenaˆq_v(a)‡r_v(a), whereq_v(a) is0 or a limit and r_v(a)5v. We s ayaisann-limitifq_v(a)>0 andr_v(a)5n 1, and other- wise, we say a is n-isolated. An n-limit a isan n;v-limit if qv(a) has countable cofinality; clearly, ifa5v1, then it isann;v-limit iff it isann- limit. Note that all ordinalsare 1-isolated andais 2-isolated iff it is isolated in the usual sense of the term.

For a function f :O ! C, we denote the support of f by supp(f), and we further let supp_I(f)ˆ fb2supp(f):b isn-isolatedg and supp_L(f)ˆ fb2supp(f):bisann-limitg. We begin with an elementary observation.

LEMMA 1.1. If V is a valuated pⁿ-socle,f ˆf_V :O ! C and b2supp_L(f),then q_v(b)is a limit point ofsupp_I(f).

PROOF. Ifdisn-isolated andd5l^defˆqv(b), then letabe the smallest ordinal such thatd5a2supp(f); soab. Ifaisann-limit, thenaˆm‡k where mˆq_v(a)>d andkˆr_v(a)5n 1. Letx2V(a)[p] V(a‡1)[p]

and find y such that jyj_Vˆm and p^kyˆx. Since k‡15n and p^k‡1yˆpxˆ0, there isaz2V(d‡1) such thatpzˆy. If a⁰ˆ jzj_V >d, thena⁰5ma, and there isaz⁰2V(a⁰‡1) such thatpz⁰ˆy. Therefore, z z⁰2V(a⁰)[p] V(a⁰‡1)[p], and sof(a⁰)6ˆ0. However, thiscontradicts the choice ofa. Soa2supp_I(f) andd5a5l, proving the result. p

We will say f :O ! Cisn-isolatedif everya2supp(f) isn-isolated; in particular, f will always be 1-isolated. The next elementary observation is a description of the Ulm functions of the class of strongly n-summable valuatedpⁿ-socles (see [4], Corollary 1.7).

LEMMA1.2. If f :O ! Cis n-isolated,then it is n-summable.

PROOF. For an n-isolated ordinal b, let kˆminfn;b‡1g and V_bˆ hx_bibe a cyclic valuated pⁿ-socle of orderp^k with jx_bj_Vb ˆ0, when b5n 1, and otherwise,jx_bj_Vbˆb (n 1). We then letVbe the valuated direct sum of f(b) copiesofV_bfor allb. It can easily be checked that f ˆf_V,

so that f isn-summable, as required. p

In particular, any function f :O ! C is 1-summable. We will say f :O ! Cisann;v-limitif it is the characteristic function of a set of the form fg_ig_i5v[ fbg, where b isann;v-limit ordinal, and g_i fori5visa strictly ascending sequence of n-isolated ordinals with limit qv(b).

Unsurprisingly, a countable valuatedpⁿ-socleVisann;v-limitif f_V isan n;v-limit function. The following result gives a concrete picture of such objects.

LEMMA1.3. If f :O ! Cisan n;v-limit function,then f isn-summable.

PROOF. Suppose, as above, f isthe characteristic function of fg_ig_i5v[ fbg; s etlˆq_v(b),kˆr_v(b). There isclearly no lossof generality in assumingg₀n 1. LetWbe generated byyandfx_ig_i5vsubject to the relationsp^k‡1yˆ0, and fori5v,pⁿxiˆy. It isstraightforward to check that a valuation onW can be defined by the formulas:jp^{`</sup>yj<sub>W</sub> ˆl‡`for 0`k, jp<sup>`}x_ij_Wˆg_i (n 1)‡` for 0`n 1, and if zˆjy‡

`<sub>1</sub>x<sub>i1</sub>‡ ‡`_mx_im, wherepⁿ6 j`<sub>1</sub>;. . .; `_i, then

jzj_W ˆminfjjyj_W;j`1xi1j<sub>W</sub>;. . .;j`mximj_Wg:

It can also be checked that ifz2pW ˆ hpx_i:i5vianda5jzj_W, then there isaz⁰2Wsuch thatpz⁰ˆzanda jz⁰j_W.

From this, it follows thatV ˆW[pⁿ] isa countable valuatedpⁿ-socle. It also follows that there is a valuated decomposition

V[p]ˆ hp^kyi M

i5v

hp^{n 1}(x_i x_i‡1)i

! :

Thisimpliesthat f_V ˆf, asrequired. p

If f :O ! Cisa function, andab 1, then letR^b

a f ˆ P

ag5bf(g):We say f iscountableifR¹

0 f @0.

THEOREM 1.4. Suppose f :O ! C is a countable function. Then the following are equivalent:

(a) f is n-summable;

(b) f is the sum of a collection of n;v-limit functions and an n-isolated function;

(c) Ifb2supp_L(f),then qv(b)is a limit point ofsupp_I(f).

PROOF. First, assuming (a), then (c) follows from Lemma 1.1.

Next, assuming (b), then (a) follows from Lemmas 1.2 and 1.3.

To complete the proof, suppose (c) holds for f; we will then verify (b).

Let Iˆ f(b;g):b2supp_L(f) andg5f(b)g;

kˆ jIj @0andf(b_j;g_j)g_j5kbe a listing ofI. For eachj5kthere isa strictly increasing sequence, fa_{j;`</sub>g<sub>`5v}ˆC_jsupp_I(f), with limitqv(b_j); we can also obviously pick these so that ifq_v(b_j)ˆq_v(b_j0), thenC_j ˆC_j⁰.

Let fN_jg_j5k be disjoint infinite subsets of v. For each j5k, we in- ductively defineT_j C_j by the equation

T_jˆ fa<sub>j;`</sub>:`2 N_jg (T₀[ [T_{j 1}):

Clearly, these sets are disjoint.

Claim:T_jis infinite. The first term is infinite, so it will suffice to show that if j⁰5j, then fa<sub>j;`</sub>:`2 N_jg \T_j⁰ isfinite. If q_v(b_j)ˆq_v(b_j0), then C_jˆC_j⁰ and clearly

fa<sub>j;`</sub>:`2 N_jg \T_j⁰ fa<sub>j;`</sub>:`2 N_jg \ fa_j⁰<sub>;`</sub>:`2 N_j⁰g ˆ ;:

On the other hand, ifq_v(b_j)6ˆq_v(b_j0), thenfa<sub>j;`</sub>:`2 N_jg \T_j⁰ C_j\C_j⁰is finite, sinceC_jandC_j⁰ have different suprema.

For every (b_j;g_j)2I, letf_jbe the characteristic function offb_jg [T_j; s o fj isann;v-limit function. In addition, ifbisann-limit ordinal, then

X

j5k

f_j

(b)ˆ jf(b_j;g_j)2I:bˆb_jgj ˆf(b):

And ifbisn-isolated, then P

j5kf_j

(b) equals1 ifb2 S

j5kT_j supp_I(f), and otherwise, it equals 0. It follows that there is ann-isolated functiongsuch that f ˆ P

j5kf_j

‡g, thereby establishing the result. p

Since a function f :O ! Cisn-summable iff it is the sum of a collection of countablen-summable functions, Theorem 1.4 immediately implies the next result.

COROLLARY1.5. A function f :O ! Cis n-summable iff it is the sum of a collection of n;v-limit functions and an n-isolated function.

COROLLARY1.6. If V is an n-summable valuated pⁿ-socle,then it is isometric to a valuated direct sumi2IVi,where each Viis either cyclic or an n;v-limit.

PROOF. By Corollary 1.5, f_V isthe sum of a collection of n;v-limit functionsand ann-isolated function, and each term is the Ulm function of a valuatedpⁿ-socle that is ann;v-limit or a valuated direct sum of cyclics. So

the result follows from Theorem 0.1. p

COROLLARY1.7. Ifais an ordinal,lˆq_v(a)and kˆr_v(a),thenais the length of some n-summable valuated pⁿ-socle V iff05k5n implies that l has countable cofinality. In fact,if l has countable cofinality or kn,then we can choose V to be countable.

PROOF. Iflhascofinalityk, then letfb_ig_i5kbe a strictly increasing set ofn-isolated ordinals with limitl. First, ifkˆ0, letKˆ fb_ig_i5k. Next, if 05k5n, thenkˆv,a 1 is ann-limit and we letKˆ fb_ig_i5v[ fa 1g.

Lastly, ifkn, thena 1ˆl‡k 1 isn-isolated and we letKˆ fa 1g.

In any of these cases, let f be the characteristic function of K. It follows from Theorem 1.5 that f ˆfVfor somen-summable valuatedpⁿ-socleV. It iseasy to check thatV will have lengtha, and that iflhascountable co- finality orkn, thenV will be countable.

Conversely, suppose l isa limit ordinal of uncountable cofinality and 05k5n. If the valuatedpⁿ-socleV iseither cyclic or an n;v-limit, then f_V(a 1)ˆ0. It followsthat there cannot be ann-summable valuatedpⁿ-

socle of lengtha. p

We now restrict our attention to functions f :v₁! C. If l isa limit ordinal, then let

f⁰(l)ˆ Z

l‡n 1

l

f and f(l)ˆinf Z^l

a

f :a5l

:

Observe thatf(l) iseither 0 or an infinite cardinal. We say f isn-thinif there is a closed and unbounded subsetCv₁consisting of limit ordinalsl such that f⁰(l)ˆ0. Further, we say f isn-admissibleif

(1.A) for every limitl5v1 we have f⁰(l)f(l); and

(1.B) either f isn-thin, orfa5v1:f(a) @1gisunbounded inv1. Any function f :v₁! Cis 1-isolated and hence 1-summable. In addition, if nˆ1, then f⁰(l)ˆ0, so f is 1-thin and 1-admissible.

We now point out that forn>1, the study ofn-summable functions can be reduced to the study of 2-summable functions, which will simplify our discussion. Ifn>1 andaisisolated, then let f⁰(a)ˆf(a‡n 2). Ifnˆ2, then f⁰ˆf.

LEMMA1.8. If f :v1! Cand n>1,then (a) f is n-isolated iff f⁰is2-isolated;

(b) f is an n;v-limit iff f⁰is a2;v-limit;

(c) f is n-admissible iff f⁰is2-admissible;

(d) f is n-summable iff f⁰ is2-summable.

PROOF. (a), (b) and (c) follow immediately from the definitions, and (d)

followsfrom (a), (b) and Corollary 1.5. p

LEMMA1.9. Suppose f :v₁! Cis2-admissible andl5v₁is a limit.

(a) If f(l)is uncountable,a5l andg5f(l),then there is an isolated ordinala⁰such that f(a⁰)is uncountable,a5a⁰5landg5f(a⁰).

(b) If f(l)6ˆ0,thenlis a limit point ofsupp_I(f).

PROOF. We verify (a), the proof of (b) being even more straightfor- ward.

Since f(l) R^l

a‡1f, we have g^0defˆ maxfg;vg5f(a⁰) for some a⁰ with a5a⁰5l. Choosea⁰to be the smallest ordinal satisfying these conditions.

We need to show thata⁰is isolated, so assume that it is actually a limit.

By (1.A),g⁰5f(a⁰)f(a⁰), sof(a⁰) isuncountable andg5f(a⁰). Arguing asin the last paragraph withlreplaced bya⁰, we haveg⁰5f(a⁰⁰) for somea⁰⁰ witha5a⁰⁰5a⁰, contradicting the minimality ofa⁰. p We now come to the main point of thissection, the characterization ofn- summable functions defined onv1.

THEOREM1.10. For a function f :v₁! C,the following are equivalent:

(a) f is n-summable;

(b) f is the sum of a collection of n;v-limit functions and an n-isolated function;

(c) f is n-admissible.

PROOF. If nˆ1 then any such function satisfies any of these condi- tions. So we may assumen>1. Replacing f by f⁰, by Lemma 1.8, we may assumenˆ2.

By Corollary 1.5, (a) and (b) are equivalent, so we need to show they are equivalent to (c). Suppose first that (b) holds, and let f ˆP

i2If_i‡g, where each f_iisa 2;v-limit andgis2-isolated. It iseasy to check that (1.A) holds forgand each f_i, which readily impliesthat it holdsfor f, aswell.

Regarding (1.B), observe first that iffa5v1:f(a) @1gisunbounded inv1, then we are clearly done; so we may assume it is bounded, say by m5v1. If (1.B) fails, then the setSˆsupp_L(f)\(m;v1) isstationary inv1. For everyl2Sthere isani_l2Isuch that f_il(l)6ˆ0. Since each f_i isa 2;v-limit function, it follows that the assignment l7!i_l isinjective. For everyl2Swe can then find anal2(m;l) such that fil(al)6ˆ0. Note that l7!alwill be a regressive function, so by Fodor's Lemma (see, for example, [12], Theorem 8.7), there isana2(m;v1) such that a_lˆa for alll in an uncountable subsetRofS. Thisimpliesthat

f(a)X

i2I

f_i(a)X

l2R

f_il(a)ˆ jRj ˆv₁:

Thiscontradictsthe fact that f(a) iscountable for alla>m. Therefore, we have shown that (b) implies (c).

Conversely, supposing (c) holds, we verify that (b) follows. LetUbe the closure offa5v₁:f(a) @₁gin the order topology. Define f_u;f_c:v₁! C by the conditionssupp(f_u)U, supp(f_c)v₁ U and f ˆf_u‡f_c. Clearly, fc(a) @0for alla5v1, and ifaisisolated, then fu(a) will be 0 or uncountable. We will be done if we can verify the following:

CLAIM1. (b) holdsfor f_u. CLAIM2. (b) holdsfor f_c. Starting with Claim 1, let

Iˆ f(b;g):b2supp_L(f_u) andg5f_u(b)ˆf(b)g:

If iˆ(b;g)2I, then clearly f(b) isuncountable. Therefore, by Lem- ma 1.9(a), there is a strictly increasing sequenceKiˆ fai;jg_j5vsupp_I(fu) with limitbsuch thatg5f(a)ˆfu(a) for alla2Ki. Let fi:v1!kbe the characteristic function offbg [K_i; s o f_iisann-limit.

We need to establish two facts:

(1) for allb2supp_L(fu), we haveP

i2If_i(b)ˆfu(b); and (2) for alla2supp_I(f_u), we haveP

i2If_i(a)f_u(a):

For (1), if b2supp_L(fu), then fi(b)ˆ1 iff iˆ(b;g), where g5fu(b).

Thishappensfor precisely fu(b) elements i2I, so that (1) follows. Re- garding (2), ifa2supp_I(fu), then fu(a) @1. Further, ifiˆ(b;g)2Iand f_i(a)ˆ1, then b>a and g5f_u(a). Since thiscan happen for at most

@₁f_u(a)ˆf_u(a) elements i2I, we can conclude that (2) follows. So we have verified Claim 1.

Turning to Claim 2, we first define a closed and unbounded subset Cv1 asfollows: If U isunbounded, then we let CˆU. On the other hand, ifU isbounded bym5v₁, then by (1.B) we can find such a subset D(m;v₁) consisting of limit ordinalslsuch that f(l)ˆ0. In thiscase, we letCˆU[D:Since supp(f_c)v₁ U and f(l)ˆ0 for alll2D, it fol- lowsthat supp(fc)v1 Cand that fc agreeswith f onv1 C.

SinceCisclosed,v1 Cis the disjoint union of sets of the form (d_i;e_i) fori2I, wheree_i5v₁. Let f_cˆP

i2If_iwhere supp(f_i)(d_i;e_i). Since f_iis countable, Claim 2, and hence the entire result, will follow once we show f_i satisfies Theorem 1.4(c) (wherenˆ2).

Letl5v1be a limit ordinal such that f_i(l)6ˆ0; we need to show thatl isa limit point of supp_I(f_i). Note thatd_i5l5e_iand by Lemma 1.9(b),lisa limit point of supp_I(f). Sincef,f_candf_iagree on (d_i;e_i), it followsthatlisa

limit point of supp_I(f_i), asrequired. p

2. Realizing Valuatedpⁿ-Socles

In this section we discuss which v1-bounded valuated pⁿ-socles V appear asthe pⁿ-socle of some group. In fact, we will start by being a bit more general. We say a valuated group V is group-like if for all ordinals a, we have (pV)(a‡1)ˆp(V(a)); in other words, if x2pV and a5jxj_V, then there isay2V such thata jyj_V and pyˆx. Note that if pⁿVˆ f0g, then pV V[p^{n 1}], which readily impliesthat any valuated pⁿ-socle is group-like. We say a valuated group V is sup- ported by a group Gif V is an essential subgroup of G so that for all x2V, jxj_V ˆ jxj_G. We will s ay V is realizable if it issupported by some group G. For valuated pⁿ-socles, this agrees with our previous terminology.

Recall from [16] that if V isa valuated group, there isa groupH(V) containingV as a nice subgroup such thatH(V)=V istotally projective. In this construction we may clearly assume thatV andH(V) have the same length.

PROPOSITION2.1. SupposeV is a valuated group. ThenV is realizable iff it is group-like and there is a balanced subgroupZH(V)such that there is a valuated decompositionH(V)[p]ˆZ[p]V[p].

PROOF. Suppose first thatV issupported by the groupG. To s how thatV isgroup-like, letx2(pV)(a‡1). Thenxˆpzfor somez2V. In addition, there isay2p^aGsuch that pyˆx. Note that p(y z)ˆ0 s o that y z2G[p]V. Therefore, yˆz‡(y z)2V\p^aGˆV(a) and pyˆx, as required.

Next, since V isa nice subgroup of H(V) and H(V)=V istotally pro- jective, by ([6], Corollary 81.4) the identity map V!V extendsto a homomorphismp:H(V)!G. It iseasy to see that the range ofpisa pure subgroup ofGcontainingV. And sinceVis essential inG, we can conclude thatpissurjective. LetZbe itskernel.

Since for allawe have (p^aG)[p]ˆV(a)[p]p((p^aH(V))[p])(p^aG)[p], it followsthatZ isbalanced in H(V). The identity map V[p](ˆG[p])! V[p](H(V)) being an isometry leads to the valuated decomposition H(V)[p]ˆZ[p]V[p], giving one implication.

Conversely, suppose V isgroup-like and we are given ZH(V) as indicated. LetGˆH(V)=Zandp:H(V)!Gbe the natural epimorphism.

It isclear thatpmapsV onto an essential subgroup ofG. We will be done if we can show that for allx2V, we havejxj_V ˆ jp(x)j_G. We certainly have jxj_V ˆ jxj_H(V) jp(x)j_G. We establish the reverse inequality by in- duction on the order ofx.

Suppose first that x2V[p] and aˆ jp(x)j_G. Since Z isa balanced subgroup ofH(V), there isay2(p^aH(V))[p] such thatp(y)ˆx. We mus t haveyˆx‡z, wherez2Z[p], so thata jyj_H jxj_V, asrequired.

Suppose now that this holds for all elements ofV[p^{k 1}], andx2V[p^k], where aˆ jp(x)j_G. Since px2V[p^{k 1}], by induction a‡1 jp(px)j_Gˆ jpxj_V. Since V isgroup-like, there isay2V(a) such thatpyˆpx. Note thatx y2V[p] andjp(x y)j_Ga, so thatjx yj_V a. However, this impliesthatjxj_V ˆ j(x y)‡yj_V a, asrequired. p PROPOSITION 2.2. If V is an v-bounded valuated group,then V is realizable iff it is group-like.

PROOF. By Proposition 2.1, if V isrealizable, then it isgroup-like.

Therefore, assumeV isv-bounded and group-like. SinceVisv-bounded, H(V) will be separable. And sinceV isnice inH(V), the quotientH(V)=V will be a separable totally projective group, i.e., it isS-cyclic. Thisimplies

that there isa valuated decompositionH(V)[p]ˆFV[p], whereFisa free,v-bounded valuated vector space (see, for example, [11], Lemma 1). It followsthat there isa pure (and hence isotype)S-cyclic subgroupZH(V) such thatZ[p]ˆF. SinceFisclosed inH(V)[p] in the inducedp-adic to- pology, it followsthatZisclosed inH(V). ThismeansthatG^defˆH(V)=Zis separable. Therefore,Zisnice, and hence balanced, in H(V). So by Pro-

position 2.1,Visrealizable. p

Since any valuated pⁿ-socle is group-like, the next statement follows directly from Proposition 2.2.

COROLLARY 2.3. If V is an v-bounded valuated pⁿ-socle,then it is realizable.

The main purpose of this section is to describe precisely when a givenn- summable valuatedpⁿ-socle is realizable. As discussed in the introduction, it is also natural to ask when a group G supported by an n-summable valuatedpⁿ-socleV isdeterminedbyV, i.e., ifG⁰isany other group sup- ported byV, thenGG⁰. If s uch aGisdetermined byV, we will sayVis uniquelyrealizable.

For example, ifV isanv-boundedn-summable valuatedpⁿ-socle, then by Corollary 2.3,V isrealizable. In fact, the summability ofV[p] implies that any group supported byVwill beS-cyclic. In other words,v-bounded n-summable valuatedpⁿ-socles are uniquely realizable. In the next section, we consider when this remains the case for groups of greater length. For now, we present a simple example.

EXAMPLE2.4. SupposeV isa valuatedpⁿ-socle which isn-summable and v2ˆv‡v-bounded. If f ˆfV and f(v) ^v2R

v‡n 1f @1, then V is realizable, but not uniquely realizable. In fact, there are groupsAandA⁰ supported byV such thatA=p^vAisS-cyclic andA⁰=p^vA⁰isnot.

PROOF. Sincefisn-admissible,f(v)f⁰(l), which readily impliesthat fis an admissible function; so there is a dsc groupAwith f ˆf_A. It follows that A[pⁿ] is n-summable, so by Theorem 0.1, there is an isometry A[pⁿ]V; in other words,Visrealizable.

On the other hand, let Mˆ fm5v:f(m) @0g, and let kn 1 be an integer such that f(v‡k) @1. LetBˆ L

m2MZ_p^m‡k‡2:Next, letXbe a pure subgroup of the torsion completionBcontainingBsuch thatX=Bhas

rank@1. It can be verified thatYˆX=B[p^k‡1] isn-summable and

f_Y(a)ˆ

1; if a2M;

@₁; if aˆv‡k;

0; otherwise.

8>

<

>:

If we letA⁰ˆAY, thenA⁰will also ben-summable. We have constructed A⁰so that f_A⁰ˆf_Aˆf. ThisimpliesthatA⁰[pⁿ] is also isometric toV.

Clearly, sinceAisa dsc group,A=p^vAisS-cyclic. On the other hand, A⁰=p^vA⁰has a summand isomorphic to

Y=p^vY ˆ(X=B[p^k‡1])=(X[p^k‡1]=B[p^k‡1])X=X[p^k‡1]p^k‡1X;

which isnotS-cyclic. It followsthatA⁰=p^vA⁰isalso notS-cyclic, so thatV

isnot uniquely realizable. p

Iflisa limit ordinal andVisa valuated group, then thel-topologyonV usesV(b) forb5las a neighborhood base of 0. Naturally, a subgroupWof Vissaid to bel-denseif it isdense in thistopology, i.e., for allb5lwe have V ˆV(b)‡W. We letLlV be the completion ofV in thel-topology, i.e., the inverse limit ofV=V(b) over b5l. Asan exception to our overall re- striction to primary groups, L_lV may have elementsof infinite order.

However, ifV isa valuatedpⁿ-socle, thenpⁿL_lV ˆ f0g. There isa natural mapV!L_lVwhose kernel isV(l), leading to an inclusionV=V(l)L_lV.

We then setPlVˆ(LlV)[p]=(V=V(l))[p] andQlV ˆLlV=(V=V(l)). IfG isa group andlhascountable cofinality,G=p^lGwill be isotype andl-dense inL_lG, andQ_lGwill be divisible.

We now review some additional concepts from [4]. Ifaisan ordinal andVis a valuatedpⁿ-socle, then a subgroupW ofV isa-highif it ismaximal with respecttoW\V(a)ˆ f0g. Ana-high subgroupWofVisn-isotypeinV(i.e.,it isa valuatedpⁿ-socle under the induced valuation), and ifaisa limit,Wisa- dense in V. If a is n-isolated, then there is a valuated decomposition VˆWU, which we refer to as astandarda-decompositionofV. If, in addition,an 1, thenUV(a n‡1). We now connect these definitions.

LEMMA2.5. Suppose V is a valuated pⁿ-socle,l5v1is a limit ordinal and W isl-high in V. Suppose further that G is a group supported by W, p:L_lG!Q_lG is the canonical epimorphism and m:V !L_lG is the natural extension of the homomorphism WG!L_lG (so that V(l)is the kernel ofm). Then there is a natural isomorphism

PlV (QlG)[p]=p(m(V))[p]:

PROOF. Note that (LlG)[pⁿ] will be complete in the (induced) l-to- pology andWV will bel-dense in it; so we can identify (LlG)[pⁿ] with L_lW L_lV. Therefore,

Pl(V) ˆ (LlV)[p]=(V=V(l))[p]

ˆ (L_lG)[p]=m(V)[p]

 [(L_lG)[p]=G[p]]=[m(V)[p]=G[p]]

 (QlG)[p]=p(m(V))[p];

completing the proof. p

We now describe a natural way to construct a group supported by a valuatedpⁿ-socle. The following is the main inductive step.

LEMMA2.6. Suppose V is a valuated pⁿ-socle,l5l⁰5v₁are limit or- dinals and W_lW_l⁰are,respectively,landl⁰-high in V. Let G_lbe a group supported by W_land A be a group supported by W_l⁰(l)(where,technically, we shift the valuation in the latter group byl). Then there is a group G_l⁰ containing Glsupported by W_l⁰such that p^lG_l⁰ ˆA iff

Z^l‡v

l‡n 1

f r(P_lV):

()

PROOF. LetAˆJK, whereJisa maximalp^{n 1}-bounded summand ofA. There isa standardl‡n 1-decompositionW_l⁰ ˆWl‡n 1Y, where Wl‡n 1isl‡n 1-high inVcontainingWl,Wl‡n 1(l)ˆJandYˆK[pⁿ].

LetDbe a divisible hull ofW_{l‡n 1}=W_landEbe a divisible hull ofK. There is an embeddingJˆW_{l‡n 1}(l)J^0defˆ[W_{l‡n 1}(l)‡W_l]=W_lD:

By a standard construction (cf., [8], Theorem 106), the existence ofG_l⁰ isequivalent to the existence of a commutative pull-back diagram

Therefore, givenG_l, the existence ofG_l⁰isequivalent to the existence of a homomorphismf:DE!Q_lG_l with kernelJ⁰K A.

The natural homomorphism m:V !L_lG_l, whose kernel is V(l), in- ducesa homomorphism g₀:Wl‡n 1=Wl!QlGl, whose kernel isJ⁰. The

divisibility of QlGl impliesthat g₀ extendsto a homomorphism g₁:D!QlGl. Since Wl‡n 1=Wl is(algebraically) the direct sum of a collection of copiesofZpⁿandp^{n 1}J⁰ˆ f0g, it followsthat (W_{l‡n 1}=W_l)=J⁰ is essential inD=J⁰. Thisimpliesthat the kernel ofg₁isalsoJ⁰. In addition, by Lemma 2.5, there is an isomorphism

P_l(V)(Q_lG)[p]=p(m(V))[p]ˆ(Q_lG)[p]=g₁(D)[p]:

A standard computation shows that the rank ofE=K agreeswith the rank of a basic subgroup of K, which is ^l‡vR

l‡n 1f. Therefore, there isa homomorphism s:E!QlGl such that fˆg₁‡s:DE!QlGl has

kernelJ⁰KAiff () holds. p

We will say the valuated group V hasan v₁-high tower if it isthe smoothly ascending union of a chain of subgroups indexed by the countable limit ordinals,fWlg_l5v1, such that for eachl,Wl isl-high inV. For ex- ample, ifV is a-bounded where a5vv, thenV clearly hasan v1-high tower. IfVisa valuatedpⁿ-socle or a group, thenV hasanv1-high tower iff the socle V[p] hasan v₁-high tower. In particular, ifV isa group or valuatedpⁿ-socle that is summable, then it has anv₁-high tower.

THEOREM2.7. Suppose the valuated pⁿ-socle V hasv1-high tower and f ˆf_V. Then V is realizable iff for every countable limit ordinall,we have

Z^l‡v

l‡n 1

f r(P_l(V)):

PROOF. Let W_l be asabove. For each limit ordinal l5v₁, we in- ductively construct an ascending chain of groupsG_lsupported byW_l. Iflis a limit of limit ordinals, we can constructGlby taking a union. Suppose then that we have constructed Wl and we want to construct Wl‡v. By Cor- ollary 2.3, we can find a groupAsupported byW_l‡v(l). So by Lemma 2.6, we can extendG_lto get the requiredG_l‡v iff our cardinality condition is

satisfied. p

We will need a pair of observations about the above construction, so assume the notation in the proof of Lemma 2.6. If ^l‡vR

l‡n 1f ˆr(P_l(V)) is infinite, then we could construct two different extensionsf;f⁰:DE! Q_lG_lofg₁, both with kernelJ⁰KA, such thatfmapsonto the torsion

subgroup of QlGl, and f⁰ doesnot. If G_l⁰ and G⁰_l0 are the groupscon- structed with these two homomorphisms, we could conclude thatQlG_l⁰ is torsion-free andQ_lG⁰_l0isnot (in other words,G_l=p^lG_lisl-torsion complete andG_l⁰=p^lG_l⁰ is not). This establishes the first part of the next result.

COROLLARY 2.8. Suppose the n-summable valuated pⁿ-socle V is realizable and f ˆfV. If either

(1) for some limit ordinall5v1, ^l‡vR

l‡n 1f ˆr(Pl(V))is infinite,or (2) f(v) R^v2

v‡n 1f @1,

then there are non-isomorphic groups G and G⁰supported by V.

PROOF. Regarding (2), let Wv2 be an v2-high subgroup of V. As in Example 2.4, we can construct non-isomorphic groupsAandA⁰of lengthv2 supported byW_v2; in particular,A=p^vAandA⁰=p^vA⁰will not be isomorphic.

If we extend these to groups G and G⁰ supported by V, then since G=p^vGA=p^vAandG⁰=p^vG⁰A⁰=p^vA⁰, we can conclude thatGandG⁰

are not isomorphic. p

We have the following immediate consequence of Theorem 2.7, which is an extension of Corollary 2.3.

COROLLARY2.9. Any v‡n 1-bounded valuated pⁿ-socle V is rea- lizable.

We now specialize this to the case of valuated pⁿ-socles that are n- summable. The next result allows us to compute the required invariants.

LEMMA2.10. If V is an n-summable valuated pⁿ-socle,f^defˆfVandlis a countable limit ordinal,then r(PlV)ˆ(f(l))^@0.

PROOF. Letjˆf(l) andb₀5lbe chosen so that R^l

b0

f ˆR^l

b f ˆjfor all b₀b5l. We may clearly assume that b₀ is n-isolated, so there is a standard b₀-decompositionVˆWU, where W isb₀-high in V. It fol- lowsthatPlV isnaturally isomorphic toPlU. ReplacingV byU, we may assumeR^l

0 f ˆj:

SinceV is a valuated direct sum of countable groups, the same will be true of V_l^defˆV=V(l), and B^defˆV_l[p]. Since a countable valuated vector

space is free, we can conclude thatBisfree. Clearly, fB(b)ˆf(b) for all b5l, and we can identify (LlV)[p] withLlB, so thatPlV PlB:

Ifa_iis a strictly increasing sequence that converges tol, then we can write Bˆ L

i5vB_i, where for each i5v we have B_i(a_i)ˆB_i and B_i(a_i‡1)ˆ f0g. It followsthatL_lBisisometric toBˆ Q

i5vB_i, so thatP_lVis isomorphic toB=B. SinceB=Bhasrankj^@0, the result follows. p Thisbringsusto the main goal of thissection, i.e., the generalization of the Existence Theorem for Principal p-Groupsfrom [9] promised in the introduction. It isan immediate consequence of Theorem 2.7, Lemma 2.10 and the fact that anv₁-boundedn-summable valuatedpⁿ-socle has anv₁- high tower.

THEOREM 2.11. Suppose V is an v₁-bounded n-summable valuated pⁿ-socle and f ˆf_V. Then V is realizable iff for every countable limit or- dinallwe have

Z^l‡v

l‡n 1

f f(l)^@0:

For example, if f(a)ˆ1 for av, f(v‡1)ˆk and f(a)ˆ0 for a>v‡1, then f is 2-admissible, so there is a valuated p²-socleV with

f ˆf_V. However, thisV isrealizable iffk2^@0.

3. Unique Realization

We say a function f :v₁! Cisv₁-countable, if f(a) iscountable for all a5v₁. The following issimilar to ([4], Corollary 1.7).

PROPOSITION3.1. SupposeV is a valuatedpⁿ-socle and f^defˆfV isv1- countable. ThenV isn-summable iff it is summable and fisn-thin.

PROOF. Suppose first thatVisn-summable, so that fisn-admissible.

It followsimmediately thatVissummable. And it followsfrom (1.B) that f isn-thin.

Conversely, supposeVissummable and f isn-thin. Leta0ˆ0 and for 05i5v1, leta_i be a strictly increasing enumeration of a CUB subset of countable limit ordinalssuch that f⁰(a_i)ˆ0 for all 05i5v₁. After col-

lecting terms, there is a valuated decomposition V[p]ˆ L

i5v1

Si; where S0V[p] is a1-high and for i>0, S_iV(a_i‡n 1)[p] is a_i‡1-high.

Clearly, eachS_iiscountable.

LetW_ibe ana_i‡1-high subgroup ofV(a_i) such thatW_i[p]ˆS_i. It fol- lowsthatWiiscountable andn-isotype inV. By ([4], Corollary 1.6), there isa valuated decompositionV ˆ L

i5v1

Wi, completing the argument. p COROLLARY3.2. Suppose G is a group such that f_Gisv₁-countable.

Then G is n-summable iff it is summable and f_Gis n-thin. In this case, G=p^aG will be countable for alla5v₁.

PROOF. The first conclusion follows by applying Proposition 3.1 to G[pⁿ]. To verify the last sentence, ifHisp^a‡v-high inG, then there isan isomorphismH=p^aHG=p^aG. SinceH[p] iscountable, so isH. Therefore,

G=p^aGisalso countable. p

For example, if f(a)ˆ1 for alla5v₁, andV isa free valuated vector space with f ˆf_V, then V isrealizable, so there isa summable group G such that f_Gˆf. SinceG=p^aGwill be countable for alla5v₁, thisGwill be a Cv1-group; thisisthe type of group constructed in [1]. Note that this group will not be 2-summable, since f isnot 2-thin. More generally, if we define fso that f(a)ˆ0 wheneveraisann-limit and f(a)ˆ1 whenevera is n-isolated, then there is an n-summable group G such that f ˆf_G. However, since f_Gisnotn‡1-thin,Gwill not ben‡1-summable.

We now turn to a discussion of when a realizablen-summable valuated pⁿ-socles is, in fact, uniquely realizable. We start with the case where the Ulm function isv1-countable.

THEOREM 3.3. Let V be a realizable n-summable valuated pⁿ-socle such that f^defˆf_V:v₁! Cisv₁-countable. Assuming the continuum hy- pothesis (CH),V is uniquely realizable iff it is countable.

PROOF. Clearly, ifV iscountable then any group supported byVwill be countable, and such groups are classified by their Ulm functions. [This direction clearly doesnot depend on CH.]

Conversely, supposeV isuncountable; i.e., supp(f) isunbounded. We need to construct non-isomorphic groupsG_S andG_Tsupported byV. Let fW_lg_l5v1be anv₁-high tower ofV. LetS_l;T_l, for limit ordinalsl5v₁, be a smoothly ascending chains of countable sets such that Sl\V ˆ Tl\V ˆWl and jSl‡v (Sl[V)j ˆ jTl‡v (Tl[V)j ˆ @0 and let S;T

be their unions. As sets, we will let GSˆSand GTˆT and we will in- ductively define group structures on Sl and Tl so that Sl[pⁿ]ˆ T_l[pⁿ]ˆW_l. It isimportant to note that sinceSvwill be pure andv-dense inS_l, ifAisa reduced group, then a homomorphismf :S_l!Aisuniquely determined by itsrestriction toS_v.

Given CH, the cardinality of the set of functions f :S_v!T is

@1@0 ˆ @1, so we can index them by fifori5v1. LetEiv1fori5vibe a collection of pairwise disjoint stationary subsets consisting of limit ordinals (see, for example, [12], Lemma 8.8).

Supposel⁰isa limit and for alll5l⁰we have defined group structures onS_landT_lsupported byW_l. Ifl⁰is a limit of limits, then we simply take unions. Suppose then, that we have constructed our group structures onSl

andT_l, andl⁰ˆl‡v. We now divide the construction into two parts.

CASE1. l2E_iand f_i(S_v)T_lextends to an isomorphismg:S_l!T_l. IdentifySlandTlusinggand call the resultGl. Then asin the proof of Lemma 2.6, there are group structures onSl‡vandTl‡v that extendGl

and are supported byW_l‡v, for whichS_l‡v=p^lS_l‡vandT_l‡v=p^lT_l‡vmap to distinct subgroups of L_lG_l. In particular, thismeansthat the iso- morphismg:S_l!T_ldoes not extend to an isomorphismS_l‡v=p^lS_l‡v! Tl‡v=p^lTl‡v.

CASE2. Case 1 does not apply.

Extend the group structures onSlandTltoSl‡vandTl‡vin any way so that they are supported byW_l‡v.

Let G_Sˆ S

l5v1

S_l andG_Tˆ S

l5v1

T_l. It followsthat V issupported by bothG_SandG_T. We claim that they are not isomorphic; assume otherwise, and let h:G_S!G_T be such an isomorphism. It follows that Cˆ fl5v₁:h(S_a)ˆT_agisclosed and unbounded inv₁. Leti5v₁be such that hrestricted toSvagreeswithfi. SinceEiisstationary, it followsthat there isanl2Ei\C. Note thathwill induce an isomorphism fromGS=p^lGS S_l‡v=p^lS_l‡v to G_T=p^lG_TT_l‡v=p^lT_l‡v, which isan extension of the isomorphismg^defˆhj_Sl:S_l!T_l. Thisiscontrary to the construction from Case 1, which shows thatGS andGT are not isomorphic. SoV isnot un-

iquely realizable. p

Observe that the continuum hypothesis is equivalent to the condition that@^@₀⁰ @₁ˆ @^‡₀:By theCEH(for ``countable exponent hypothesis) we

will mean the statements thatk^@0k^‡for all infinite cardinalsk. Clearly, the generalized continuum hypothesis implies the CEH. The following result, therefore, generalizes Theorem 3.3.

THEOREM 3.4. Suppose V is a realizable n-summable valuated pⁿ- socle and f ˆf_V. If V(v‡n 1)is countable,then V is uniquely realiz- able. In fact,any group supported by V must be a dsc group.

Conversely,assuming the CEH,if V(v‡n 1)is uncountable,then V is not uniquely realizable.

PROOF. Suppos e firs t thatV(v‡n 1) iscountable andGisa group supported byV. LetHbep^{v‡n 1}-high inG. SinceGisn-summable, by ([4], Theorem 3.5),Hmust be a dsc group. Sincer(G=H)ˆr(V(v‡n 1)) @₀, it followsfrom Wallace'sTheorem (see, for example, [3], Proposition 1.1) that Gisa dsc group. Therefore,Gisdetermined byV.

Conversely, supposeV(v‡n 1) isuncountable. Letkˆ ^v2R

v‡n 1f.

CASE1. kisuncountable.

Letmˆf(v). Ifkm, then it followsfrom Corollary 2.8(2) thatVisnot uniquely realizable. We may therefore assume thatk>m, so that the CEH implieskm^@0. By Theorem 2.11,km^@0, so thatkˆm^@0ˆr(Pv(V)). In thiscase, Corollary 2.8(1) impliesthatV isnot uniquely realizable.

CASE2. kˆ @0.

Subcase 2a: For allawithv‡n 1a5v₁we have f(a) @₀.

It followsfrom Theorem 3.3 that there are non-isomorphic groupsA0

and A1 supported by V(v‡n 1) (where, again, we shift values by v‡n 1). These two groups are contained in groupsG1andG2supported byV, whereA₀ˆp^{v‡n 1}G₁ andA₁ˆp^{v‡n 1}G₂. SinceG₁andG₂will also not be isomorphic, it follows thatV isnot uniquely realizable.

Subcase 2b: For someawithv‡n 1a5v₁we have f(a) @₁. Let av be chosen minimally so that f(a) @1. Since kˆ R

v2

v‡n 1f ˆ @0, we can conclude thatav2. Since f isn-admissible,amust be n-isolated. Ifqv(a)ˆlv2, thenf(l) R^l

v‡n 1f ˆ @0. In addition, by Theorem 2.11

@₁ Z^l‡v

l‡n 1

f r(P_l(V))ˆf(l)^@0 @^@₀⁰ ˆ @₁:

It followsfrom Corollary 2.8(1) thatVisnot uniquely realizable, completing

the proof. p

Again, assuming the CEH, the last result states that ann-summable groupGisuniquely determined by f_Giff it isa dsc group andp^{v‡n 1}Gis countable. We now consider the particular case wherenˆ1.

A groupGissaid to bev-totallyS-cyclicif every separable subgroup of GisS-cyclic. These groups were studied in [2], where, for example, it was shown that they are precisely the dsc groupsGfor whichp^vGiscountable.

Therefore, we have the following consequence of Theorem 3.4 fornˆ1.

COROLLARY 3.5. Assuming the CEH,a summable group G is un- iquely determined by f_Giff it isv-totallyS-cyclic.

For example, if f(a)ˆ @1for allavandf(a)ˆ0 fora>v, then asin Example 2.4, there are summable groupsAand A⁰ with Ulm function f such thatAisa dsc group andA⁰isnot. So, asa summable groupAisnot uniquely determined by f. However by Theorem 3.4, asa 2-summable groupA isuniquely determined by f.

In ([4], Theorem 3.8), it wasshown that ifGisa group andG[pⁿ] isn- summable for all positive integers n, then G must be a dsc group. We conclude this paper with an analogous result for Ulm functions that illus- trates the power of our realization results.

THEOREM 3.6. A function f :v₁! C is admissible iff it is n-ad- missible for all positive integers n.

PROOF. If fis admissible, then there is a dsc groupGsuch thatf ˆf_G. It followsthatG[pⁿ] isn-summable andf ˆf_G[pⁿ_]. So by Theorem 1.10,fis n-admissible for each positive integern.

Conversely, suppose for each positive integern, f isn-admissible. So by Theorem 1.10, there isann-summable valuatedpⁿ-socleVn such that f_Vnˆf. It isclear thatV_n‡1[pⁿ] will also be ann-summable valuatedpⁿ- socle and that f_Vn‡1[pⁿ_] ˆf, so that by Theorem 0.1 there is an isometry V_nV_n‡1[pⁿ]. If we identify these two, then we can defineG to be the union S

1n5vVn. Note that G hasa valuation j j_G determined by the valuationson the variousVn. In addition, if x2G and a5jxj_G, then x2Vn for somen, and so x2Vn‡1[pⁿ]. Hence, there isay2Vn‡1G such that pyˆx and a jyj_G. Thismeansthat for all x2G, jxj_Gˆ supfjyj_G‡1:y2Gandpyˆxg. An easy induction then implies thatj j_G

isthe height function onG. SinceG[pⁿ]ˆVnisn-summable for eachn, it followsfrom ([4], Theorem 3.8) that G isa dsc group. Therefore, f ˆf_G

will be admissible. p

The last result could be proven combinatorially using the definition (1.A and B), but the argument is less intuitive and significantly longer than the above.

REFERENCES

[1] D. CUTLER,Another summable CV-group, Proc. Amer. Math. Soc.,26(1970), pp. 43-44.

[2] P. DANCHEV- P. KEEF,An application of set theory to v‡n-totally p^v‡n- Projective primary abelian groups, Mediterr. J. Math. (to appear).

[3] P. DANCHEV- P. KEEF,Generalized Wallace theorems, Math. Scand.,104(1) (2009), pp. 33-50.

[4] P. DANCHEV- P. KEEF,n-Summable valuated pⁿ-socles and primary abelian groups, Commun. Algebra,38(2010), pp. 3137-3153.

[5] P. DANCHEV- P. KEEF,Nice elongations of primary abelian groups, Publ.

Mat.,54(2) (2010), pp. 317-339.

[6] L. FUCHS,Infinite Abelian Groups,Volumes I & II, Academic Press, (New York, 1970 and 1973).

[7] L. FUCHS,Vector spaces with valuations, J. Algebra,35(1975), pp. 23-38.

[8] P. GRIFFITH,Infinite Abelian Group Theory, The University of Chicago Press (Chicago and London, 1970).

[9] K. HONDA, Realism in the theory of abelian groups III, Comment. Math.

Univ. St. Pauli,12(1964), pp. 75-111.

[10] R. HUNTER - F. RICHMAN- E. WALKER, Existence Theorems for Warfield Groups, Trans. Amer. Math. Soc.,235(1978), pp. 345-362.

[11] J. IRWIN- P. KEEF,Primary abelian groups and direct sums of cyclics, J.

Algebra,159(2) (1993), pp. 387-399.

[12] T. JECH, Set Theory (Third Millennium Edition), Springer (Berlin, 2002).

[13] P. KEEF, On v1-p^v‡n-projective primary abelian groups, J. Algebra and Number Theory Academia,1(1) (2010), pp. 53-87.

[14] P. KEEF - P. DANCHEV, On m;n-balanced projective and m;n-totally pro- jective primary abelian groups, Submitted.

[15] P. KEEF- P. DANCHEV,On n-simply presented primary abelian groups, To appear in Houston J. Math.

[16] F. RICHMAN- E. WALKER,Valuated groups, J. Algebra,56(1) (1979), pp. 145- 167.

Manoscritto pervenuto in redazione il 3 gennaio 2011.