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 pjn ∈ 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 j by any power of p since xk = p xj+k. Yet another description of Z/p∞ is the multiplicative group of all p-power roots of unity. These are complex numbers of the form e2πin/pk where n is an integer. An isomorphism φ with the additive version above is given by

2πi/pk φ(xk) = e . 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 coor- dinate 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).

1 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

2 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, y0 are two elements of G so that ny = x = ny0 then y − y0 is n-torsion so must be zero.) a Consequently, there is an action of Q on G given by taking b 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

3 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.