Finite Groups, Torsion and Locally Finite

Torsion and Locally Finite

Consistent with modules over a pid, a torsion element has finite order and a torsion group has all torsion elements. A torsion free group has no torsion elements.

In a locally finite group, the subgroup generated by any finite set of elements is finite. The polynomials over a finite field form a locally finite group under addition, even though the entire group is infinite.

It is not known whether torsion groups are always locally finite. If not, restrict atention to that finite set of generators that produces an infinite group. Each generator, and each element spanned, has finite order. Abelian groups are out, since generators can be clumped together, and the size of the group is bounded by the product of the orders of the generators.