Abelian Groups - László Fuchs [Springer]

54 2 Direct Sums and Direct Products
(13) (C. Walker) Generalize Theorem 2.5 to larger cardinalities : summands of
direct sums of -generated groups are of the same kind.
(14) A supplement subgroup S to some C < A is defined to be minimal with
respect to the property A D C C S. S has this property if and only if S \ C is
superfluous in A. [Hint: use the modular law in both directions.]
3 Pull-Back and Push-Out Diagrams
Pull-Backs With the aid of direct sums, we can describe two important methods
in constructing certain commutative diagrams.
Theorem 3.1. Given the homomorphisms ˛ W A ! C and ˇ W B ! C, there exists
a group G, unique up to isomorphism, along with homomorphisms W G ! A; ıW
G ! B, such that the diagram
γ
G −−−−→ A
⏐ ⏐
δ↓
↓α
B
β
−−−−→ C
is commutative, and if
G ′ γ ′
−−−−→ A
⏐
⏐ ⏐↓
↓δ ′
α
B
β
−−−−→ C
is any commutative diagram, then there exists a unique homomorphism W G 0 ! G
such that D 0 and ı D ı 0 .
Proof. Given ˛; ˇ,defineG as the subgroup of the direct sum A˚B consisting of all
pairs .a; b/.a 2 A; b 2 B/ such that ˛a D ˇb,andlet W .a; b/ 7! a; ıW .a; b/ 7! b.
This makes the first diagram commutative.
If the second diagram is commutative, then define W G 0 ! G as g 0 D
. 0 g 0 ;ı 0 g 0 / for g 0 2 G 0 ;here. 0 g 0 ;ı 0 g 0 / 2 G, since˛ 0 D ˇı 0 . Evidently, g 0 D
0 g 0 and ıg 0 D ı 0 g 0 for every g 0 2 G 0 . It is easy to see that Ker D .0; Ker ˇ/ and
Ker ı D .Ker ˛; 0/. Therefore, if 0 W G 0 ! G also satisfies 0 D 0 ;ı 0 D ı 0 ,
then . 0 / D 0 D ı. 0 /,andsoIm. 0 / Ker \ Ker ı D 0: Hence
0 D 0, thus is unique.
The uniqueness of G can be verified by considering a G 0 with the same properties.
Then by what has already been shown, there are unique maps W G ! G 0 ;
0 W G 0 ! G with the indicated properties. Then 0 W G ! G is a unique map