Subgroups of Bounded Abelian Groups
Valuated groups are a topic of central interest in Abelian group theory. On one hand, they provide a viewpoint for classical Abelian theory problems, and on the other hand are of interest in their own right. In this latter regard, there has been some prog
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New Mexico State University, Las Cruces N.M. 0. Introduction. Abelian
group
Valuated groups are a topic of central interest in On
theory.
one
hand,
they
provide
a
viewpoint
for
classical Abelian theory problems, and on the other hand are of interest in their own right.
In this latter regard, there has been some progress
in getting structure theorems for certain valuated groups. of
invariants
valuated
have
been
provided
for
finite
direct
Complete·sets
sums
·p-groups (HRWl), for finite simply presented valuated
of
cyclic p-groups
[AHW], and for direct sums of torsion-free cyclic valuated groups [AHW]. A general discussion of simply presented valuated toward a structure theory, is presented in (HW]. general
study
of
finite
valuated
p-groups
In [BHW), a basis for a is
structurE' theories for simply presented valuated valuated
p-groups, with an aim
suggested.
p-groups and for finite
p-groups are only in the initial stages.
This paper is concerned with the structure of with
a
However,
finite
bound
on
the
values
1
that
may
be
valuated
assumed
These authors were supported by NSF grant UMCS 8301606.
R. Göbel et al. (eds.), Abelian Groups and Modules © Springer-Verlag Wien 1984
by
p-groups non-zero
R. Hunter- F. Richman- E. Walker
18 elements.
Each bounded abelian group decomposes
cyclic groups of prime power order,
into a direct sum of
the number of cyclic groups of each
order in the decomposition being an invariant of the group.
Thus the
structure theory of bounded abelian groups may be considered complete. The
picture
is
far
murkier
when we attempt
bounded groups, that is, pairs
B.
bounded subgroup of to groups that are p-group
valuated group
n
A(n)
,.
is bounded and
is an ordinal, let
A
A(a)
A(a)
If
A
{a e A
•
a, denoted
is of length
p.
0,
that is,
A
is a
.\(A)
..
is a valuated
v(a)
a,
The study of pairs
.. 0.
those
with
A
.\(A) ~ n.
a A
A
The
~a}.
if
is equivalent to the study of valuated groups
0
=
B
With no loss of generality we restrict ourselves
smallest ordinal such that p B
where
p-primary for a fixed prime
a
and
Ac B
to classify subgroups of
is the
c 8
with
such that
This equivalence is made
precise in Theorem 1.2. The valuated groups of length
~
1
consist of
and these are direct sums of cyclic groups. groups of length sums of cyclics groups of length
~
2
3
The structure of valuated
is not obvious, although finite ones are direct
[ HRWl, ~
p-bounded groups,
Theorem 4).
It
turns out
that
the valuated
are direct sums of cyclics, and one of our main
theorems (Theorem 4.2) classifies those valuated groups of length To this end we construct, for each finite cyclic valuated group such that if
functor
A
is a direct sum of copies of then
A
also enable us
to
a subgroup of a valuated group only if
An
is a summand of
B
4.
C,
a
A
is
if and
FC(B) • 0 (Theorem 2.5).
The functors vAlmot:Pii
8,
C, and
~
show
that
any
2
p -bounded
Proun of finite length is a d
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