By Laszlo Fuchs
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Additional info for Infinite Abelian Groups, Volume 1 (Pure & Applied Mathematics)
Example 3. For a positive integer 11, let the functor M. : d + sd assign to A its subgroup nA, and to a : A + B the induced homomorphism a I nA : nA + nB. Example 4 . , groups G satisfying nG = 0. A functor d + d , is obtained by assigning A [ n ] to A and a 1 A [ n ] to a : A + B. I. 26 PRELIMINARIES Example 5. If %? is the category of torsion-free groups, then the function assigning to A EI the quotient group A / T ( A ) and to a : A + B in d the induced homomorphism a* : a T ( A ) H aa T ( B ) [which map is independent of the choice of a in its coset mod T ( A ) ] of A / T ( A ) into B/T(B)is a functor on d to V.
6. (a) Show that the group of all complex roots of unity is locally cyclic. 4. MODULES 19 (b) Every subgroup and every quotient group of a locally cyclic group is again locally cyclic. 7. Prove the isomorphisms: Q / Q , g Z(p">, for k = 1, 2, . P'"'/Z z Z(p">, Jp/pkJp Z(pk) 8. (L. Rtdei) If a group A contains subgroups isomorphic to any one of Z(pk)with a fixed prime p and k = 1,2, * , but n o proper subgroup of A has this property, then A r Z(p"). ] 9. I n a cyclic group A , two subgroups B and C coincide if A / B E A/C.
Let a E A = B 0 C, and write u = b + Bik)= 0, + c with b E B, c E C. The maps n : a ~ b and 8 : a ~ c are immediately seen to be epimorphisms of A onto B and C, respectively. Since nb = b, nc = 0, 8c = c, 86 = 0, and nu + 8a = a, the endomorphisms n, 8 of A satisfy: (1) n2 = 71, e2 = 0, 8n=718=0, 7 1 + 8 = l A . If we mean by a projection an idempo tent endomorphism and by orthogonal endomorphisms those with 0 products [in both orders], then (1) may be expressed by saying that a direct decomposition A = B Q C dejnes a pair of orthogonal projections with sum 1,.