All groups in this chapter will be additive. Thus extensions of A by C can be written as short exact sequences:
f g 0 → A −→ B −→ C → 0 which are sequences of homomorphisms between additive groups so that im f = ker g, ker f = 0 (f is a monomorphism) and coker g = 0 (g is an epimorphism).
Definition 17.1. The torsion subgroup tG of an additive group G is defined to be the group of all elements of G of finite order. [Note that if x, y ∈ G have order n, m then
nm(x + y) = nmx + nmy = 0 so tG ≤ G.] G is called a torsion group if tG = G. It is called torsion-free if tG = 0.
Lemma 17.2. G/tG is torsion-free for any additive group G.
Proof. This is obvious. If this were not true there would be an element of G/tG of finite order, say x + tG with order n. Then nx + tG = tG implies that nx ∈ tG so there is an m > 0 so that m(nx) = 0. But then x has finite order and x + tG is zero in G/tG.
Lemma 17.3. Any homomorphism f : G → H sends tG into tH, i.e., t is a functor.
Proof. If x ∈ tG then nx = 0 for some n > 0 so nf(x) = f(nx) = 0 which implies that f(x) ∈ tH.
Lemma 17.4. The functor t commutes with direct sum, i.e., M M t Gα = tGα.
This is obvious. The only surprise is that this does not hold for direct products. [See example 17.6 below.]
Theorem 17.5. 1. Every additive group G is an extension of a torsion subgroup by a torsion-free quotient group:
0 → T → G → F → 0.
2. Any direct sum of torsion groups is torsion.
3. Any direct product of torsion-free groups is torsion-free.
1 4. Any quotient of a torsion group is torsion.
5. If T is a torsion group and F is torsion-free then Hom(T,F ) = 0.
Proof. (1) follows from Lemma 17.2. (2) follows from Lemma 17.4. The remainingQ three properties all follow from the functorality of t: (3) t ( Fα) = 0 since it projects to tFα = 0 for all α. (4) The image of G = tG lies in tH. (5) The image of T = tT in F lies in tF = 0. We say that x ∈ G is divisible (in G) by a positive integer n if there exists an element y ∈ G so that ny = x. For example in the finite p-group Z/pk, there are no nontrivial elements divisible by pk [it cannot be divided by p more than k − 1 times] but every element is divisible by any positive integer n which is relatively prime to p. We use divisibility to examine the following example which serves two purposes. It shows that a product of torsion groups may not be torsion. [Whereas a direct sum of torsion groups is always torsion.] And it show that the canonical extension tG → G → G/tG is not always split.
Example 17.6. Let G be the direct product of the cyclic groups Z/p2k for k ≥ 1. Then G is not a torsion group since it contains elements of infinite order such as (1, 1, 1, ··· ). This group also has the property that its torsion subgroup tG is not a direct summand. To see this note first that every nonzero x ∈ G can only be divided by 2k p finitely many times. [If xk 6= 0 then x is not divisible by p .] However, G/tG contains nonzero elements which are divisible by pj for all j > 0. One 2 3 k 2k such element is the image in G/tG of z = (p, p , p , ··· ), i.e., zk = p ∈ Z/p for each k ≥ 1. For any positive integer j let y ∈ G be given by ( pk−j if k ≥ j, yk = 0 otherwise.
Then z = pjy in G/tG since the difference:
z − pjy = (p, p2, ··· , p2j−2, 0, 0, ··· ) has order p2j−2 and is thus torsion.
Definition 17.7. If G is an additive group and p is a prime, let Gp be the subgroup of all elements of G of p-power order. If G = Gp we say that G is a p-primary group.
Proposition 17.8. Any homomorphism G → H sends Gp into Hp, i.e., Gp is a functor of G.
2 Theorem 17.9 (Primary decomposition). Any torsion group is a direct sum of its primary subgroups: M G = Gp.
k1 k2 kr Proof. Suppose that x ∈ G has order n = p1 p2 ··· pr . Then the integers
n k1 k2 cki kr ni = = p1 p2 ··· pi ··· pr pki have the propertyP that their greatest common divisor is 1 so there exist integers ai so that aini = 1. But then X X x = ainix ∈ Gp P ki since nix has order pi . Thus G = Gp. This sum must be a direct sum since elements of p-power order cannot be a sum of elements of order prime to p. [See Proposition 17.11 below.]
Independent Sets Given any additive group G let G[p] be the subgroup of all x ∈ G so that px = 0. Then G[p] is a vector space over the field Z/p and therefore has a basis X. This basis has the property that it is independent in the following sense.
Definition 17.10. A set X of nonzero elements of anP additive group G is called independent if there are no nonzero finite sums mixi where mi ∈ Z unless mixi = 0 for each i. This is the same as saying that M hXi = hxi . x∈X For example:
Proposition 17.11. If the elements of a subset X of G have finite orders which are pairwise relatively prime then X is independent.
Pn Proof. Suppose that X is not independent. Then there is a relation i=1 mixi = 0 where each mixi 6= 0 and n ≥ 2 is minimal. Let a be the order of xn. Then Xn−1 amixi = 0 i=1 but each term amixi is nonzero contradicting the minimality of n.
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