Algebra I: Section 6. The structure of groups. 6.1 Direct products of ...

However, in dealing with groups we only care whether they are isomorphic. It happens that
Z3 × Z5 ∼ = Z5 × Z3 even though these groups are not “identical.”
6.1.5 Exercise. Let A1, A2, . . . be groups. Prove that the following product groups are isomorphic.
(a) A1 × A2 ∼ = A2 × A1
(b) A1 × (A2 × A3) ∼ = (A1 × A2) × A3 ∼ = A1 × A2 × A3 (as defined in (1))
(c) If one of the groups is trivial we get
A1 × (e) ∼ = A1 (e) × A2 ∼ = A2 �
In essence, the operation of forming the direct product of two groups is commutative and
associative, and the trivial group E = (e) acts as an “identity element.” The significance of (b)
is that, up to isomorphism, we get the same group if we multiply groups together all at once,
as in (1), or multiply them successively two at a time.
6.1.6 Exercise. In the product group G = A×A the diagonal is the subset ∆ = {(a, a) : a ∈ A}
(a) Show that ∆ is a subgroup of G
(b) Show that ∆ ∼ = G.
(c) Find a complete set of coset representatives for the coset space G/∆.
In general ∆ is not a normal subgroup of G ×G. Can you see why? In (c) you are looking for a
set X ⊆ G that meets each coset g∆ in a unique point. Such a set is referred to as a transversal
for the space of cosets G/∆. �
6.1.7 Example. Euclidean space R n equipped with vector addition (+) as a binary operation
is an abelian group. Its elements are defined as n-tuples x = (x1, . . . , xn) of real numbers,
elements of the Cartesian product set R × . . . × R. Comparison of the sum
x + y = (x1 + y1, . . .,xn + yn)
with (1) shows that G = (R n , +) is precisely the direct product group (R × . . . × R, +) made
up of n copies of the real line (R, +). �
6.1.8 Exercise. The set of integral vectors in R n
Z n = {x ∈ R n : x = (x1, . . . , xn) with xi ∈ Z for all i}
is a group under vector addition (+). Explain why (Z n , +) ∼ = Z × . . . × Z. If {ui} is an R-basis
for R n and
Λ = {a1u1 + . . . + anun : ai ∈ Z} = Zu1 + . . . + Zun
prove that (Λ, +) is an abelian group isomorphic to (Z n , +). What map ψ : Z n → Λ effects the
isomorphism? �
There is an important “internal” version of the direct product construction. Given a group
G and subgroups Ai ⊆ G we would like to know when G is isomorphic to the direct product
A1 × . . . × An. From (2) we can read out some obvious necessary conditions,
(i) Each Ai must be a normal subgroup in G.
(3A) (ii) The Ai must generate G in the sense that G is equal to the product set
A1 · · ·An = {x1 · · · xn : xi ∈ Ai}
Furthermore it is evident from the definition of the direct product G = A1 ×...×An that each
group element g has a unique decomposition as a product g = x1 · · ·xn with xi ∈ Ai. That
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