Torsion group
From Encyclopedia of Mathematics
(Redirected from Periodic group)
2020 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.
Comments
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.
References
[Ha] | P. Hall, "The theory of groups", Macmillan (1959) pp. Chapt. 10 MR0103215 Zbl 0084.02202 |
[Ku] | A.G. Kurosh, "The theory of groups", 1–2, Chelsea (1955–1956) (Translated from Russian) MR0071422 Zbl 0111.02502 |
[Ro] | D.J.S. Robinson, "Finiteness condition and generalized soluble groups", I, Springer (1972) MR0332990 MR0332989 |
How to Cite This Entry:
Periodic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Periodic_group&oldid=25720
Periodic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Periodic_group&oldid=25720