Download 18 Divisible groups

18
Divisible groups
Every group splits as a direct sum
G=D⊕R
where D is divisible and R is reduced. However, the divisible subgroup is
natural whereas the reduced subgroup is not. [Recall that all groups are
abelian in this chapter.]
Definition 18.1. A group G is called divisible if for every x ∈ G and every
positive integer n there is a y ∈ G so that ny = x, i.e., every element of G is
divisible by every positive integer.
For example the additive group of rational numbers Q and real numbers
R and any vector space over Q or R are divisible but Z is not divisible. [In
fact there are no nontrivial finitely generated divisible groups.]
Another example of a divisible group is Z/p∞ . This is usually defined as
a direct limit:
Z/p∞ = lim Z/pk
→
where Z/pk maps to Z/pk+1 by multiplication by p. This is the quotient
of the direct sum ⊕Z/pk by the relation that n ∈ Z/pk is identified with
pj n ∈ Z/pj+k . Another way to say this is that Z/p∞ is generated by elements
x0 , x1 , x2 , x3 , · · · which are related by xk = pxk+1 and x0 = 0. Since xk has
order pk it is divisible by any number relatively prime to p and is divisible
by any power of p since xk = pj xj+k .
Yet another description of Z/p∞ is the multiplicative group of all p-power
k
roots of unity. These are complex numbers of the form e2πin/p where n is an
integer. An isomorphism φ with the additive version above is given by
k
φ(xk ) = e2πi/p .
Divisible group satisfy properties similar to those of torsion groups:
Proposition 18.2.
1. Every product of divisible groups is divisible.
2. Every direct sum of divisible groups is divisible.
3. Every quotient of a divisible group is divisible.
Q
Proof. Let x = (xα ) ∈
Gα where each Gα is divisible. Then each coordinate xα ∈ Gα is divisible by any given n so there exists yα ∈ Gα so that
nyα = xα . Then n(yα ) = (nyα ) = x. This proves (1). To prove (2) note that
if almost all coordinates xα are zero we can choose the corresponding yα to
be zero making y ∈ ⊕Gα .
To prove that the image f (G) of a divisible group G is divisible note
that for any y = f (x) and n > 0 there is a z ∈ G so that nz = x so
y = f (nz) = nf (z) is divisible in f (G).
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Definition 18.3. For any group G let dG be the subgroup of G generated
by all divisible subgroups of G. If dG = 0 then G is called reduced.
As a consequence of the second and third statements in Proposition 18.2
we get the following.
Corollary 18.4. dG is a divisible subgroup of G and any homomorphism
G → H sends dG into dH, i.e., d is a functor.
The main property of divisible groups is that they are injective. To prove
this we need the following lemma.
Lemma 18.5 (Pasting lemma). If A, B are subgroups of C and f : A → G,
g : B → G are homomorphisms which agree on A ∩ B (i.e., f |A ∩ B =
g|A ∩ B), then there is a unique extension f + g of f and g to A + B given
by
(f + g)(a + b) = f (a) + g(b).
Proof. There is a short exact sequence
j
∆
0→A∩B −
→A⊕B →
− A+B →0
where ∆, j are given by ∆(x) = (x, −x) and j(a, b) = a + b. The condition
on f, g is equivalent to saying that ∆(A ∩ B) lies in the kernel of
f ⊕ g : A ⊕ B → G.
Consequently, we get an induced map on the quotient: f +g : A+B → G.
Theorem 18.6. A group G is divisible if and only if it satisfies the following
“injectivity” condition: Any homomorphism f : A → G from any group A
into G extends to any group B which contains A, i.e., there exists a homomorphism f : B → G so that f |A = f .
Proof. We use Zorn’s Lemma. Consider the partially ordered set P of all
pairs (C, g) where C is a subgroup of B containing A and g : C → G is an
extension of f . Let (C, g) ≤ (D, h) if C ≤ D and g = h|C.
The set P is nonempty since it contains (A, f ). Also any tower (Cα , gα )
in P has an upper bound (∪Cα , ∪gα ) ∈ P . Consequently, by Zorn’s Lemma,
P has a maximal element, say (C, g). We claim that C = B and g is the
desired extension of f to B. To see this suppose C < B. Then there exists
an x ∈ B so that x ∈
/ C. There are two cases. Either x + C has finite order
in B/C or it has infinite order.
In the second case, hC, xi = C ⊕ hxi so g : C → G can be extended to
g ⊕ 0 : C ⊕ hxi → G contradicting the maximality of (C, g).
In the first case, let n be the order of x + C in B/C, i.e., n > 0 is smallest
positive integer so that nx ∈ C. Since G is divisible there is a z ∈ G so
that nz = g(nx). By the Pasting lemma, g : C → G can be extended to the
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homomorphism g + h : C + hxi → G where h : hxi → G is given by h(x) = z.
This contradicts the maximality of (C, g). Therefore, divisible groups are
injective.
Conversely, if G is injective then it is obviously divisible: given any x ∈ G
and n > 0 let f : nZ → G be given by f (n) = x. If G is injective this extends
to a homomorphism f : Z → G. But then nf (1) = f (n) = x.
Corollary 18.7. Any divisible subgroup of G splits, i.e., if D ≤ G is divisible
then D has a complement H so that G = D ⊕ H.
Proof. Since D is divisible, it is injective. Thus the identity mapping D → D
extends to a homomorphism r : G → D. Then H = ker r is a complement
for D in G.
Corollary 18.8. Every group G splits as a direct sum
G = dG ⊕ R
where dG is the divisible subgroup of G and R is a (noncanonical) reduced
subgroup of G.
Proof. By Corollary 18.7, dG has a complement H. We just need to show
that H is reduced. But dH ≤ H ≤ G is a divisible subgroup of G and is
therefore contained in dG. So dH ≤ H ∩ dG = 0 and H is reduced.
Lemma 18.9. The torsion subgroup tD of any divisible group D is divisible,
D/tD is a vector space over Q and D ∼
= tD ⊕ D/tD.
Proof. Any x ∈ tD is divisible by n > 0 in D. So there is a y ∈ D s.t.
ny = x. But then mny = mx = 0 for some m so y is torsion and lies in tD.
Since tD is torsion it has a complement isomorphic to D/tD.
Since G = D/tD is a quotient of a divisible group it is also divisible.
Since G is also torsion-free it is uniquely divisible (if x ∈ G and y, y 0 are two
elements of G so that ny = x = ny 0 then y − y 0 is n-torsion so must be zero.)
Consequently, there is an action of Q on G given by taking ab x to be the
unique element y ∈ G so that by = ax. Thus G is a vector space over Q.
Theorem 18.10 (Classification of divisible groups). Divisible groups
are direct sums of the groups Q and Z/p∞ . Furthermore, two divisible groups
G, H are isomorphic iff
1. dimQ (G/tG) = dimQ (H/tH) and
2. dimGF (p) G[p] = dimGF (p) H[p] for all primes p.
Proof. Given any divisible group D, D = tD⊕D/tD. By the lemma, D/tD is
a direct sum of many copies of the divisible group Q. The torsion subgroup
tD is a direct sum of primary subgroups: tD = ⊕tDp . Since each tDp is
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a quotient of D, it is also divisible. Thus it remains to show that every
p-primary divisible group P is a direct sum of copies of Z/p∞ .
Let {bα }α∈I be a basis for the Z/p-vector space P [p]. Construct a direct
sum V = ⊕I Z/p∞ of copies of Z/p∞ one for every element of the index set
I. Then V [p] is a direct sum of copies of Z/p, one for each element of I so it
has a Z/p-basis {cα }α∈I indexed by I. Thus there is a (unique) isomorphism
of Z/p-vector spaces
φ : P [p] → V [p]
sending each bα to cα . Since V is injective, this extends to a homomorphism
φ : P → V . Since φ is a monomorphism on P [p] it is a monomorphism. [The
kernel K of φ must be zero since K[p] = K ∩ P [p] = 0.] The image is thus
an injective subgroup of V which must also split: V = φ(P ) ⊕ H. But the
image of φ contains V [p] so H[p] = H ∩ V [p] = 0 implies H = 0 and φ is an
isomorphism P ∼
= V . The theorem follows.
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