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ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА
2018 Математика и механика № 51
УДК 512.541.6 MSC 20K10
DOI 10.17223/19988621/51/2
Peter V. Danchev
A NOTE ON ESSENTIALLY INDECOMPOSABLE m-SUMMABLE ABELIAN ^-GROUPS
For each natural n we prove that there exists an unbounded n-summable abelian p-group which is essentially indecomposable. This example parallels a well-known result of this kind established for separable abelian p-groups.
Keywords: summable groups, essentially indecomposable groups, admissible Ulm functions, direct sums of countable groups.
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 <raj; 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: ra1 ^ 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: raj ^ 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< y with limit y, the inequality
X[ 1 ) f ^Zm ) f) holds.
It can be proved that a function f: raj ^ 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 ra-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 ra1 -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
A note on essentially indecomposable n-summable Abelian p-groups
17
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 : ra1 ^ 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[ffl + n 1 ® ) f is countable.
Proof: Suppose first that V[ffl+n ® ) 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[ffl+n® ) 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[ffl+n — ® ) 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
f (ra+m) is uncountable. In addition, the admissibility of f implies that for every P<ra, V n f is uncountable, so there is an unbounded subset S era such that for
all Pe S , f (P) is infinite. We define
if PeS; if P = ra + m; otherwise.
Since supp(h) = S U {ra+ m} e In , it is clear that h is n-admissible, so there is an n-summable group H with fH = f . Note that h is not admissible, so that H is not a direct sum of countable groups.
Since f is n-realizable, there is an n-summable group G' with fG, = f. If
G = G' © H , then G is n-summable, and since it is easy to check that f=f + h, it follows that fG = fG- + fH = f + h = f . On the other hand, since H fails to be a direct sum of countable groups, G is not a direct sum of countable groups, either. ■
REFERENCES
1. Fuchs L. (1970, 1973) Infinite Abelian Groups. Vol. I , II. New York; London: Academic Press.
2. Danchev P.V. , Keef P.W. (2010) n-Summable valuated pn-socles and primary abelian groups. Commun. Algebra. 38(9). pp. 3137-3153.
3. Keef P.W. (2011) Realization theorems for valuated pn-socles. Rend. Sem. Mat. Univ. Padova. 126. pp.151-173.
4. Danchev P.V. (2012). Valuated /"-socles and Cx n-summable abelian p-groups. Pioneer J. Math. and Math. Sci. 6(2). pp. 233-249.
5. Danchev P.V. (2013) A note on pM+n+2-projective efi n-summable abelian /-groups. Adv. Appl. Math. Sci. 12(3). pp. 151-155.
6. Danchev P.V., Keef P.W. (2009) Generalized Wallace theorems. Math. Scand. 104(1). pp. 33-50.
Received: November 6, 2017.
Peter V. DANCHEV (Professor, Department of Mathematics, Plovdiv University "P. Hilendarski", Plovdiv 4000, Bulgaria)
E-mail: [email protected]
Данчев П.В. (2018) ЗАМЕЧАНИЕ О СУЩЕСТВЕННО НЕРАЗЛОЖИМЫХ n-СУММИ-РУЕМЫХ АБЕЛЕВЫХ p-ГРУППАХ. Вестник Томского государственного университета. Математика и механика. № 51. С. 15-18
DOI 10.17223/19988621/51/2
Для каждого натурального n мы доказываем, что существует неограниченная n-суммируемая абелева р-группа, которая существенно неразложима. Этот пример параллелен известному аналогичному результату, установленному для сепарабельных абелевых р-групп.
Ключевые слова: суммируемые группы, существенно неразложимые группы, допустимые функции Ульма, прямые суммы счетных групп.
Danchev P.V. A NOTE ON ESSENTIALLY INDECOMPOSABLE n-SUMMABLE ABELIAN p-GROUPS. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika [Tomsk State University Journal of Mathematics and Mechanics]. 51. pp. 15-18
AMS Mathematical Subject Classification: 20K10.
