A finite rank torsion free abelian group
G
G
is almost completely decomposable if there exists a completely decomposable subgroup
C
C
with finite index in
G
G
. The minimum of
[
G
:
C
]
[G:C]
over all completely decomposable subgroups
C
C
of
G
G
is denoted by
i
(
G
)
i(G)
. An almost completely decomposable group
G
G
has, up to isomorphism, only finitely many summands. If
i
(
G
)
i(G)
is a prime power, then the rank 1 summands in any decomposition of
G
G
as a direct sum of indecomposable groups are uniquely determined. If
G
G
and
H
H
are almost completely decomposable groups, then the following statements are equivalent: (i)
G
⊕
L
≈
H
⊕
L
G \oplus L \approx H \oplus L
for some finite rank torsion free abelian group
L
L
. (ii)
i
(
G
)
=
i
(
H
)
i(G) = i(H)
and
H
H
contains a subgroup
G
′
G’
isomorphic to
G
G
such that
[
H
:
G
′
]
[H:G’]
is finite and prime to
i
(
G
)
i(G)
. (iii)
G
⊕
L
≈
H
⊕
L
G \oplus L \approx H \oplus L
where
L
L
is isomorphic to a completely decomposable subgroup with finite index in
G
G
.