Let S be a nonempty set of groups, and let G s be the group

corresponding to the membersof S. If(G,_{ps})and(G0,{ps0})are two direct

products, then the homomorphisms ps : G → Gs and p0s : G0 → Gs are onto

Gs, there exists a unique homomorphism8:G0→Gsuch thatps0 = ps◦8for

alls _∈S, and8is an isomorphism.

PROOF. In Figure 4.3 let H = G0 _andϕ

s = ps0. If 8 : G0 → G is the

ps◦8= ps0for alls. Reversing the roles ofGandG0, we obtain a homomorphism

80_{:G →G}0_withp0

s◦80 = psfor alls. Thereforeps◦(8◦80)= p0s◦80= ps.

In Figure 4.3 we next letH =G andϕs = ps for alls. Then the identity 1G

onGhas the same propertyps◦1G = psrelative to allpsthat8◦80has, and the

uniqueness says that8_◦80 _{=1G. Reversing the roles ofG andG}0_{, we obtain} 80_◦8₌1G0. Therefore8is an isomorphism.

For uniqueness suppose that9 : G0 _{→ G is another homomorphism with}

p0

s = ps◦9for alls ∈S. Then the argument of the previous paragraph shows

that80_◦9 ₌1G0. Applying8on the left gives9 =(8◦80)◦9 =8◦(80◦9)=

8_◦1G0 =8. Thus9=8.

Finally we have to show that the sth _{mapping of a direct product is onto}

Gs. It is enough to show that ps0 is ontoGs. Taking G as the external direct

productQ_s∈SGswithpsequal to the coordinate mapping, form the isomorphism

80 _{:G →G}0_{that has just been proved to exist. This satisfiesps = p}0

s◦80for

alls ∈S. Sinceps is ontoGs,ps0 must be ontoGs. §

Let us turn to direct sums. Part of what we seek is a definition that allows for an abstract characterization of direct sums in the spirit of Proposition 4.16. In particular, the interaction with homomorphisms is to be central to the dis- cussion. In the case of two factors, we usei1 andi2 rather than p1and p2. If

ϕ1:G1→ H andϕ2:G2→ H are two homomorphisms, then the correspond-

ing homomorphismϕofG1⊕G2toH is to satisfyϕ1=ϕ◦i1andϕ2=ϕ◦i2.

WithG1⊕G2defined, as expected, to be the same group asG1×G2, we are led

to the formula

ϕ(g1,g2)=ϕ(g1,1)ϕ(1,g2)=ϕ1(g1)ϕ2(g2).

The images of commuting elements under a homomorphism have to commute, and henceHhad better be abelian. Then in order to have an analog of Proposition 4.16, we will want to specializeH at some point toG1⊕G2, and thereforeG1

andG2had better be abelian. With these observations in place, we are ready for

the general definition.

LetSbe an arbitrary nonempty set ofabeliangroups, and letGs be the group

corresponding to the membersofS. We shall use additive notation for the group operation in eachGs. Theexternal direct sumof theGs’s consists of an abelian

groupL_s∈SGsand a system of group homomorphismsisfors ∈S. The group is

the subgroup ofQ_s∈SGsof all elements that are equal to 0 in all but finitely many

coordinates. The group homomorphisms are the mappingsis0: Gs0 → L

s∈SGs

carrying a membergs0 ofGs0 to the element that isgs0 in coordinates0and is 0 at all other coordinates. The individual groups are called thesummands, and a direct sum ofnabelian groups may be written asG1⊕ · · · ⊕Gn. The group

L

s∈SGshas theuniversal mapping propertydescribed in Proposition 4.17 and

Proposition 4.17(universal mapping property of external direct sum). Let {Gs | s ∈ S} be a nonempty set of abelian groups, and let Ls∈SGs be the

external direct sum, the associated group homomorphisms being the embedding mappingsis0 :Gs0 →

L

s∈SGs. IfH is any abelian group and{ϕs |s ∈S}is a

system of group homomorphismsϕs :Gs → H, then there exists a unique group

homomorphismϕ:L_s∈SGs → H such thatϕ◦is0 =ϕs0 for alls0∈S. Gs0 ϕs −−−→ H is0   y L s∈SGs ϕ

FIGURE4.4. Universal mapping property of an external direct sum of abelian groups.

PROOF. Existence ofϕis proved by takingϕ°_{gs}s∈S¢=Psϕs(gs). The sum

on the right side is meaningful since the element{gs}s∈S of the direct sum has

only finitely many nonzero coordinates. SinceH is abelian, the computation

ϕ°_{gs}s∈S¢+ϕ°{gs0}s∈S¢=Psϕs(gs)+Psϕs(gs0)

=Ps(ϕs(gs)+ϕs(gs0))=

P

sϕs(gs +gs0)

=ϕ°_{gs+g0s}s∈S¢=ϕ°{gs}s∈S+ {gs0}s∈S¢

shows thatϕ is a homomorphism. Ifgs0 is given and{gs}s∈S denotes the el- ement that is gs0 in the s0

th _{coordinate and is 0 elsewhere, thenϕ(i}

s0(gs0)) =

ϕ°_{gs}s∈S¢= Psϕs(gs), and the right side equalsϕs0(gs0)sincegs = 0 for all others’s. Thusϕ_◦is0 =ϕs0.

For uniqueness letϕ0:L_s∈SGs → Hbe a homomorphism withϕ0◦is0 =ϕs0 for alls0∈S. Then the value ofϕ0is determined at all elements ofLs∈SGsthat

are 0 in all but one coordinate. Since the most general member ofL_s∈SGs is a

finite sum of such elements,ϕ0is determined on all ofL_s∈SGs. §

Now we give an abstract definition of direct sum that allows for the possibility that the direct sum is “internal” in the sense that the various constituents are identified as subgroups of a given group. Again the definition is by means of a universal mapping property and will be seen to characterize the direct sum up to canonical isomorphism. LetSbe an arbitrary nonempty set ofabeliangroups, and letGs be the group corresponding to the members ofS. Adirect sumof

theGs’s consists of an abelian groupG and a system of group homomorphisms

is :Gs →Gfors ∈Swith the followinguniversal mapping property: when-

ϕs : Gs → H, then there exists a unique group homomorphismϕ : G → H

such thatϕ_◦is =ϕs for alls ∈S. Proposition 4.17 proves existence of a direct

sum, and the next proposition addresses uniqueness. A direct sum isinternalif eachGs is a subgroup ofGand each mappingisis the inclusion mapping.

Gs ϕs −−−→ H is   y G ϕ

FIGURE4.5. Universal mapping property of a direct sum of abelian groups.