A note on essentially indecomposable n-summable Abelian p-groups.pdf 0. Introduction and Fundamentals Without any exceptions, the term "group" will mean an abelian p-group, where p is a prime fixed for the duration of the paper. Our terminology and notation will be based upon [1]. In particular, if G is a group and a is an arbitrary ordinal, then paG = {x e G: htG (x) > a}, and we shall say G is separable if p®G = {0}. Likewise, for every positive integer n, the symbol G[pn ] = {g e G: png = 0} denotes the pn socle of G which can be viewed as a valuated group by consulting with [2]. About the notions of valuated pn -socles, valuated groups and their closely related specifications, we refer the interested reader to [2] and [3]. The other specific concepts will be defined below explicitly as follows: • Mimicking [2], a group G is said to be n-summable if G[pn ] decomposes as (is isometric to) the valuated direct sum of a collection of countable valuated groups (each of which will also be a valuated pn -socle). Naturally, a group G is n-summable if G[pn] is n-summable as a valuated pn -socle. Note that an n-summable group has to be summable (since a countable valuated vector space is necessarily free), and so p®1 G = {0} (see, e.g., Theorem 84.3 of [1]). In [3] was constructed for any natural n an n-summable group G which need not be n+1-summable such that G / paG is a direct sum of countable groups for all a < ю^ thus this G is not a direct sum of countable groups. • (Folklore) A group Z is said to be essentially indecomposable if whenever Z = X © Y for some groups X and Y, then either X or Y is bounded. • Imitating [3], the function f: i® ^ C is called n-realizable, provided f = fV for some n-summable valuated pn -socle V, where fV designates the Ulm function of V. In particular, considering groups, f = fG for some n-summable group G, where V = G[ pn ]. • Imitating [3], the function f: ^ C is called n-admissible, provided it is n-closed and either uncountably unbounded or n-small and, in addition, for every pair of countable ordinals p< у with limit y, the inequality X[ 1 ) f ^(Xrn ) f ) holds. + n -1, y + ra)J Y) / It can be proved that a function f: ^ C is n-admissible if, and only if, it is n-realizable (cf. [3]). The motivation for writing this short article is to promote some new ideas concerning certain indecomposable properties of n-summable groups related to valuated groups and valuated pn -socles (see, for more account, [4] and [5] too). 1. Examples and Assertions If A is any separable group, B is a basic subgroup of A and G = A / B[pn ], then the purity of B in A implies that there is an isomorphism G[ pn ] = (A[ pn ]/ B[ pn ]) © (B[ p2n ]/ B[ pn ]). Because B is ю-dense in A, it follows that the first term in this sum is praG . Considering multiplication by pn : B ^ pnB , it follows that the second term is isometric to pnB[ pn ] using the regular height function. It follows that G[ pn ] is n-summable and hence G is n-summable appealing to [2]. Note also that the isomorphism G / praG = pnA holds. An example of an essentially indecomposable separable group Z can be constructed using Corollary 76.4 of [1]. So, we come to the following: Example 1.1 There is an n-summable group G that is essentially indecomposable. Proof: If Z is a separable essentially indecomposable group and A is a separable group such that pnA = Z , then let B be a basic subgroup of A and let G = A / B[ pn ], so that G[pn ] is n-summable. If G = X © Y, then Z = pnA = G / praG = (X / praX)© (y / praY). Therefore, either ( X / pra X ) or else (Y / praY ) is bounded, so that either X or Y is bounded, which implies that G is also essentially indecomposable. ■ In other words, a group can have only inessential decompositions and still have a pn -socle which splits into an infinite number of countable valuated summands. In spite of the parallel between direct sums of countable groups and -bounded n-summable valuated pn -socles, there are many n-summable groups that are not direct sums of countable groups. In fact, we have the following construction: Example 1.2. Any n-summable group G is a summand of a group with an admissible Ulm function that is not a direct sum of countable groups. Proof: We can construct a direct sum of countable groups H which is large enough so that the Ulm function of T = G © H is admissible. This means that there is a direct sum of countable groups T' such that T and T' have the same Ulm functions. Since both T[pn ] and T' [pn ] are n-summable, they are isometric. On the other hand, T is not a direct sum of countable groups since this would imply that so is G - a contradiction. ■ Again, this shows that an n-summable group with the same pn -socle as a direct sum of countable groups need not be a direct sum of countable groups. The next result characterizes the Ulm functions for which such a phenomenon can occur. The following statement can also be deduced directly from results presented in [3], but we here give a more transparent proof, however. Theorem 1.3. Suppose f : ю1 ^ C is n-realizable. Then every n-summable group G with fG = f is a direct sum of countable groups if, and only if, V[© + n 1 © ) f is countable. Proof: Suppose first that V[©+n - ffl ) f is countable, and let H be p©+n -1 -high in G. Since G is n-summable, by Theorem 3.5 of [2], H must be a direct sum of countable groups. Since r(G/H) = V[©+n-i ffl ) f - ^o, it follows from Wallace's theorem (see, for instance, Proposition 1.1 of [6]) that G is a direct sum of countable groups. Conversely, suppose V[©+n -1 © ) f is uncountable; our aim is then to produce an n-summable group G with fG = f which fails to be a direct sum of countable groups. If f is not admissible, then any n-summable group G with fG = f will fail to be a direct sum of countable groups, so we may assume that f is admissible. In particular, we can conclude that V r , f is uncountable, so there is an integer m > n -1 such that ^[©+n-1, ©-2)y ' b f (ю+m) is uncountable. In addition, the admissibility of f implies that for every Р

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