In the classification of finite Abelian groups, we define a subset Sp of a group G$,$ for each prime p$,$ consisting of all elements whose order is a power of p$.$ We show that Sp is a subgroup using the argument that if g$,$ h have orders pi$,$ pj $,$ then pi$+$j $($g $+$ h$)=$ pi$+$jg $+$ pi$+$jh $=0.$What property$/$properties of the group G are being used here, and how are they used?