An abelian group that is neither periodic nor torsion-free is called mixed. If is an abelian group and is its torsion subgroup, then the factor group is torsion-free. However, in general the torsion subgroup is not a direct summand of , so is not isomorphic to . Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group of integers is torsion-free -module.Lal, R., Algebra 2: Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier (Berlin, Heidelberg: Springer, 2017), p. 206.
An abelian group that is neither periodic nor torsion-free is called mixed. If is an abelian group and is its torsion subgroup, then the factor group is torsion-free. However, in general the torsion subgroup is not a direct summand of , so is not isomorphic to . Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. The additive group of integers is torsion-free -module.Lal, R., Algebra 2: Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier (Berlin, Heidelberg: Springer, 2017), p. 206.