Let
A
=
(
A
1
,
…
,
A
n
)
A= ({A_1},\ldots ,{A_n})
be an
n
n
-tuple of subgroups of the additive group,
Q
Q
, of rational numbers and let
G
(
A
)
G(A)
be the kernel of the summation map
A
1
⊕
⋯
⊕
A
n
→
∑
A
i
{A_1} \oplus \cdots \oplus {A_n} \to \sum \;{A_i}
and
G
[
A
]
G[A]
the cokernel of the diagonal embedding
∩
A
1
→
A
1
⊕
⋯
⊕
A
n
\cap \,{A_1} \to {A_1} \oplus \cdots \oplus {A_n}
. A complete set of isomorphism invariants for all strongly indecomposable abelian groups of the form
G
(
A
)
G(A)
, respectively,
G
[
A
]
G[A]
, is given. These invariants are then extended to complete sets of isomorphism invariants for direct sums of such groups and for a class of mixed abelian groups properly containing the class of Warfield groups.