Abstract Algebra: Theory and Applications · Problem 31

Permutation Groups — Thomas Judson

Mathematics — Abstract Algebra · cyclic-groups-torsion-subgroup

Problem

Let

G

be an abelian group. Show that the elements of finite order in

G

form a subgroup. This subgroup is called the torsion subgroup of

G

. Hint. The identity element in any group has finite order. Let

g, h \in G

have orders

m

and

n

, respectively. Since

(g^{-1})^m = e

and

(gh)^{mn} = e

, the elements of finite order in

G

form a subgroup of

G