Editing Torsion subgroup (section)

==Examples and further results==
[[Image:Lattice torsion points.svg|right|thumb|200px|The 4-torsion subgroup of the quotient group of the complex numbers under addition by a lattice.]]

*The torsion subset of a non-abelian group is not, in general, a subgroup. For example, in the [[infinite dihedral group]], which has [[presentation of a group|presentation]]:

:  <math>\langle x,y \mid x^2=y^2=1 \rangle </math>

:the element ''xy'' is a product of two torsion elements, but has infinite order.
* The torsion elements in a [[nilpotent group]] form a [[normal subgroup]].<ref>See Epstein & Cannon (1992) [https://books.google.com/books?id=DQ84QlTr-EgC&pg=PA167 p. 167]</ref>
*Every finite abelian group is a torsion group. Not every torsion group is finite however: consider the direct sum of a [[countably infinite|countable]] number of copies of the [[cyclic group]] ''C''<sub>2</sub>; this is a torsion group since every element has order 2. Nor need there be an upper bound on the orders of elements in a torsion group if it isn't [[generating set of a group|finitely generated]], as the example of the [[factor group]] '''Q'''/'''Z''' shows.
*Every [[free abelian group]] is torsion-free, but the converse is not true, as is shown by the additive group of the [[rational number]]s '''Q'''.
*Even if ''A'' is not finitely generated, the ''size'' of its torsion-free part is uniquely determined, as is explained in more detail in the article on [[rank of an abelian group]].
*An abelian group ''A'' is torsion-free [[if and only if]] it is [[flat module|flat]] as a '''Z'''-[[module (mathematics)|module]], which means that whenever ''C'' is a subgroup of some abelian group ''B'', then the natural map from the [[tensor product of abelian groups|tensor product]] ''C'' ⊗ ''A'' to ''B'' ⊗ ''A'' is [[injective]].
*Tensoring an abelian group ''A'' with '''Q''' (or any [[divisible group]]) kills torsion. That is, if ''T'' is a torsion group then ''T'' ⊗ '''Q''' = 0. For a general abelian group ''A'' with torsion subgroup ''T'' one has ''A'' ⊗ '''Q''' ≅ ''A''/''T'' ⊗ '''Q'''.
*Taking the torsion subgroup makes torsion abelian groups into a [[coreflective subcategory]] of abelian groups, while taking the quotient by the torsion subgroup makes torsion-free abelian groups into a [[reflective subcategory]].