Could we finite order elements in an infinite group and expl

Could we finite order elements in an infinite group and explain the reason with step by step solution

Solution

G=(Z/2Z)G=(Z/2Z), or indeed HH for any finite group HH.

Let HH be a finite group, and let G=HG=H, the set of infinite sequences of element of HH with multiplication defined componentwise. If the order of HH is nn, then clearly gn=1Ggn=1G for each gGgG.

Added: For a more interesting example, let Gn=Z/nZGn=Z/nZ for nZ+nZ+, and let GG be the direct sum of the GnGn’s. In other words, GG is the set of sequences

mk:kZ+kZ+Gkmk:kZ+kZ+Gk

such that only finitely many mkmk are non-zero. Then GG is infinite, all of its elements have finite order, and for each nZ+nZ+ GG has an element of order nn.