Difference between revisions of "Torsion group"
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| − | A [[Group|group]] in which every element has finite order. Any | + | {{MSC|20}} |
| + | {{TEX|done}} | ||
| + | |||
| + | A ''torsion group'' (also called ''periodic group'') | ||
| + | is a | ||
| + | [[Group|group]] in which every element has finite order. Any torsion | ||
| + | [[Abelian group|Abelian group]] splits into a direct sum of primary groups with respect to distinct prime numbers. See | ||
| + | [[Burnside problem|Burnside problem]] on torsion groups for finiteness conditions of torsion groups. | ||
====Comments==== | ====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==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Ha}}||valign="top"| P. Hall, "The theory of groups", Macmillan (1959) pp. Chapt. 10 {{MR|0103215}} {{ZBL|0084.02202}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ku}}||valign="top"| A.G. Kurosh, "The theory of groups", '''1–2''', Chelsea (1955–1956) (Translated from Russian) {{MR|0071422}} {{ZBL|0111.02502}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ro}}||valign="top"| D.J.S. Robinson, "Finiteness condition and generalized soluble groups", '''I''', Springer (1972) {{MR|0332990}} {{MR|0332989}} | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 21:20, 29 April 2012
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:
Torsion group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_group&oldid=11928
Torsion group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion_group&oldid=11928
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article