196 3. PRODUCTS OF GROUPS (a) G is a direct product of cyclic groups, G Š Z a 1 ± Z a 2 ± ²²² ± Z a s ± Z k ; where a i ³ 2 , and a i divides a j for i ´ j . (b) The decomposition in part (a) is unique, in the following sense: If G Š Z b 1 ± Z b 2 ± ²²² ± Z b t ± Z ` ; where b j ³ 2 , and b i divides b j for i ´ j , then ` D k , s D t , and a j D b j for all j . An element of ﬁnite order in an abelian group G is also called a torsion element . It is easy to see that an integer linear combination of torsion ele-ments is a torsion element, so the set of torsion elements forms a subgroup, the torsion subgroup G tor . We say that G is a torsion group if G D G tor and that G is torsion free if G tor D f0 g . It is easy to see that G=G tor is torsion free. See Exercise 3.6.1 . An abelian group is ﬁnite if, and only if, it is a ﬁnitely generated tor-sion group. (See Exercise 3.6.4 .) Note that if G is ﬁnite, then ann .G/ D f r 2 Z W rx D0 for all x 2 G g is a nonzero subgroup of Z (Exercise 3.6.5 ). Deﬁne the period of G to be the least positive element of ann .G/ . If a is the period of G , then ann .G/ D a Z , since a subgroup of Z is always generated by its least positive element.
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.