Groups and Group Actions
Theorem 4.14 (Second Isomorphism Theorem) Let H 1 and H 2 be subgroups
of a groupGwithH2normal inG. ThenH1∩H2is a normal subgroup ofH1, the
setH1H2of products is a subgroup ofG withH2as a normal subgroup, and the
maph1(H1∩H2)7→h1H2is a well-defined canonical isomorphism of groups
H1/(H1∩H2)=∼(H1H2)/H2.
PROOF. The setH1∩H2is a subgroup, being the intersection of two subgroups.
Forh1inH1, we haveh1(H1∩H2)h1−1⊆h1H1h−11⊆H1sinceH1is a subgroup
andh1(H1∩H2)h−11 ⊆ h1H2h−11 ⊆ H2 since H2 is normal inG. Therefore
h1(H1∩H2)h−11⊆ H1∩H2, andH1∩H2is normal inH1.
The setH1H2of products is a subgroup sinceh1h2h01h02=h1h01(h01−1h2h01)h02
and since(h1h2)−1 = h−11(h1h2−1h−11), and H2 is normal in H1H2since H2is
normal inG.
The functionϕ(h1(H1∩H2))=h1H2is well defined since H1∩H2 ⊆ H2,
andϕrespects products. The domain ofϕis{h1(H1∩H2)|h1∈ H1}, and the
kernel is the subset of this such thath1lies inH2as well asH1. For this to happen,
h1must be inH1∩H2, and thus the kernel is the identity coset ofH1/(H1∩H2).
Henceϕis one-one.
To see thatϕis onto(H1H2)/H2, let h1h2H2be given. Thenh1(H1∩H2)
maps toh1H2, which equalsh1h2H2. Henceϕis onto. §
3. Direct Products and Direct Sums
We return to the matter of direct products and direct sums of groups, direct products having been discussed briefly in Section 1. In a footnote in Section II.4 we mentioned a general principle in algebra that “whenever a new systematic con- struction appears for the objects under study, it is well to look for a corresponding construction with the functions relating these new objects.” This principle will be made more precise in Section 11 of the present chapter with the aid of the language of “categories” and “functors.”
Another principle that will be relevant for us is that constructions in one context in algebra often recur, sometimes in slightly different guise, in other contexts. One example of the operation of this principle occurs with quotients. The construction and properties of the quotient of a vector space by a vector subspace, as in Section II.5, is analogous in this sense to the construction and properties of the quotient of a group by anormalsubgroup, as in Section 2 in the present chapter. The need for the subgroup to be normal is an example of what is meant by “slightly different guise.” Anyway, this principle too will be made more precise in Section 11 of the present chapter using the language of categories and functors.
Let us proceed with an awareness of both these principles in connection with direct products and direct sums of groups, looking for analogies with what hap- pened for vector spaces and expecting our work to involve constructions with homomorphisms as well as with groups.
The external direct productG1×G2was defined as a group in Section 1 to
be the set-theoretic product with coordinate-by-coordinate multiplication. There are four homomorphisms of interest connected withG1×G2, namely
i1:G1→G1×G2 given by i1(g1)=(g1,1),
i2:G2→G1×G2 given by i2(g2)=(1,g2),
p1:G1×G2→G1 given by p1(g1,g2)=g1,
p2:G1×G2→G2 given by p2(g1,g2)=g2.
Recall from the discussion before Proposition 4.5 that Proposition 2.30 for the direct product of two vector spaces does not translate directly into an analog for the direct product of groups; instead that proposition is replaced by Proposition 4.5, which involves some condition of commutativity.
Warned by this anomaly, let us work with mappings rather than with groups and subgroups, and let us use mappings in formulating a definition of the direct product of groups. As with the direct product of two vector spaces, the mappings to use are p1 and p2but not i1 andi2. The way in which p1 and p2enter is
through the effect of the direct product on homomorphisms. Ifϕ1 : H → G1
andϕ2 : H → G2 are two homomorphisms, thenh 7→ (ϕ1(h), ϕ2(h))is the
corresponding homomorphism ofHintoG1×G2. In order to state matters fully,
let us give the definition with an arbitrary number of factors.
LetS be an arbitrary nonempty set of groups, and letGs be the group cor-
responding to the members of S. The external direct product of the Gs’s
consists of a group Qs∈SGs and a system of group homomorphisms. The
group as a set is
×
s∈SGs, whose elements are arbitrary functions from S toS
s∈SGs such that the value of the function ats is inGs, and the group law is
°
{gs}s∈S¢°{g0s}s∈S¢= {gsg0s}s∈S. The group homomorphisms are the coordinate
mappingsps0 : Q
s∈SGs →Gs0with ps0 °
{gs}s∈S¢=gs0. The individual groups Gs are called thefactors, and a direct product ofn groups may be written as
G1×· · ·×Gninstead of with the symbolQ. The groupQs∈SGshas theuniversal
mapping propertydescribed in Proposition 4.15 and pictured in Figure 4.2.
Proposition 4.15(universal mapping property of external direct product). Let {Gs |s∈S}be a nonempty set of groups, and letQs∈SGs be the external direct
product, the associated group homomorphisms being the coordinate mappings ps0 :
Q
s∈SGs → Gs0. If H is any group and{ϕs |s ∈ S}is a system of group homomorphismsϕs : H →Gs, then there exists a unique group homomorphism
Gs0 ϕs0 √−−− H ps0 x Q s∈SGs ϕ
FIGURE4.2. Universal mapping property of an external direct product of groups. PROOF. Existence ofϕis proved by takingϕ(h)= {ϕs(h)}s∈S. Thenps0(ϕ(h)) = ps0
°
{ϕs(h)}s∈S¢=ϕs0(h)as required. For uniqueness letϕ0 : H → Q
s∈SGs
be a homomorphism withps0◦ϕ0=ϕs0for alls0∈S. For eachhin H, we can writeϕ0(h) = {ϕ0(h)s}s∈S. Fors0in S, we then haveϕs0(h)=(ps0 ◦ϕ0)(h) = ps0(ϕ0(h))=ϕ0(h)s0, and we conclude thatϕ0=ϕ. § Now we give an abstract definition of direct product that allows for the possi- bility that the direct product is “internal” in the sense that the various factors are identified as subgroups of a given group. The definition is by means of the above universal mapping property and will be seen to characterize the direct product up to canonical isomorphism. LetSbe an arbitrary nonempty set of groups, and let Gs be the group corresponding to the membersof S. Adirect productof the
Gs’s consists of a groupGand a system of group homomorphismsps :G →Gs
fors ∈ S with the followinguniversal mapping property: whenever H is a group and{ϕs |s ∈S}is a system of group homomorphismsϕs : H →Gs, then
there exists a unique group homomorphismϕ : H → G such that ps ◦ϕ =ϕs
for alls∈S. Proposition 4.15 proves existence of a direct product, and the next proposition addresses uniqueness. A direct product isinternalif each Gs is a
subgroup ofG and each restrictionpsØØG
s is the identity map. Gs ϕs √−−− H ps x G ϕ
FIGURE4.3. Universal mapping property of a direct product of groups.