Torsion group

2010 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

A torsion group (also called periodic group) is a group in which every element has finite order. Any torsion Abelian group splits into a direct sum of primary groups with respect to distinct prime numbers. See Burnside problem on torsion groups for finiteness conditions of torsion groups.

For any group $G$, its torsion subgroup is defined by $T(G) = \{ g\in G \;|\; \exists n\in\N \textrm{ such that } g^n = e\}$. It is a normal subgroup and the quotient $F(G) = G/T(G)$ is the torsion-free quotient group of $G$. Both $T(\;.\;)$ and $F(\;.\;)$ are functors.