17 Torsion subgroup tG 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 2 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 2 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 2 tG then nx = 0 for some n > 0 so nf(x) = f(nx) = 0 which implies that f(x) 2 tH. Lemma 17.4. The functor t commutes with direct sum, i.e., M M t G® = tG®: 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 ( F®) = 0 since it projects to tF® = 0 for all ®. (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 2 G is divisible (in G) by a positive integer n if there exists an element y 2 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 2 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 2 Z=p for each k ¸ 1. For any positive integer j let y 2 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 2 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 2 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 2 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 2 Z unless mixi = 0 for each i. This is the same as saying that M hXi = hxi : x2X 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. 3.