1963. PRODUCTS OF GROUPS(a)Gis a direct product of cyclic groups,GŠZa1Za2ZasZk;whereai2, andaidividesajforij.(b)The decomposition in part (a) is unique, in the following sense:IfGŠZb1Zb2ZbtZ`;wherebj2, andbidividesbjforij, then`Dk,sDt,andajDbjfor allj.An element of finite order in an abelian groupGis also called atorsionelement. 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 subgroupGtor. We say thatGis atorsion groupifGDGtorand thatGistorsion freeifGtorD f0g. It is easy to see thatG=Gtoristorsion free. See Exercise3.6.1.An abelian group is finite if, and only if, it is a finitely generated tor-sion group. (See Exercise3.6.4.) Note that ifGis finite, then ann.G/Dfr2ZWrxD0for allx2Ggis a nonzero subgroup ofZ(Exercise3.6.5). Define theperiodofGto be the least positive element of ann.G/.Ifais the period ofG, then ann.G/DaZ, since a subgroup ofZis alwaysgenerated by its least positive element.
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