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Difference between revisions of "Period of a group"

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The least common multiple of the orders of the elements in the group (it is assumed that the group is periodic and that the set of orders of all its elements is bounded). The period of a group is also called the exponent of the group.
 
The least common multiple of the orders of the elements in the group (it is assumed that the group is periodic and that the set of orders of all its elements is bounded). The period of a group is also called the exponent of the group.
  
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The order of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072160/p0721601.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072160/p0721602.png" /> is the least positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072160/p0721603.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072160/p0721604.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072160/p0721605.png" /> is the indentity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072160/p0721606.png" />).
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The order of an element $x$ of a group $G$ is the least positive integer $n$ such that $x^n=e$ (where $e$ is the indentity of $G$).
  
 
Cf. also [[Periodic group|Periodic group]].
 
Cf. also [[Periodic group|Periodic group]].

Latest revision as of 12:02, 23 August 2014

The least common multiple of the orders of the elements in the group (it is assumed that the group is periodic and that the set of orders of all its elements is bounded). The period of a group is also called the exponent of the group.


Comments

The order of an element $x$ of a group $G$ is the least positive integer $n$ such that $x^n=e$ (where $e$ is the indentity of $G$).

Cf. also Periodic group.

How to Cite This Entry:
Period of a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Period_of_a_group&oldid=33107
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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