Difference between revisions of "Order (group theory)"

m
Line 1: Line 1:
 
In group theory, the term '''order''' has different meanings in different contexts.
 
In group theory, the term '''order''' has different meanings in different contexts.
  
The order of a group <math>G</math>, sometimes denoted <math>\ord{G}</math>, is the [[cardinality]] of its underlying [[set]].
+
The order of a group <math>G</math>, sometimes denoted <cmath>\ord{G}</cmath>, is the [[cardinality]] of its underlying [[set]].
  
 
The order of an element <math>x</math> of <math>G</math>, <math>\text{ord}(x)</math>, is the order of the [[subset]] generated by <math>x</math>.  If <math>\text{ord}(x)</math> is finite, then it is also the least positive integer <math>n</math> for which <math>x^n=e</math>.
 
The order of an element <math>x</math> of <math>G</math>, <math>\text{ord}(x)</math>, is the order of the [[subset]] generated by <math>x</math>.  If <math>\text{ord}(x)</math> is finite, then it is also the least positive integer <math>n</math> for which <math>x^n=e</math>.

Revision as of 18:16, 28 March 2018

In group theory, the term order has different meanings in different contexts.

The order of a group $G$, sometimes denoted

\[\ord{G}\] (Error compiling LaTeX. Unknown error_msg)

, is the cardinality of its underlying set.

The order of an element $x$ of $G$, $\text{ord}(x)$, is the order of the subset generated by $x$. If $\text{ord}(x)$ is finite, then it is also the least positive integer $n$ for which $x^n=e$.

In number theory, for $a$ relatively prime to $n$, the order of $a$ (mod $n$) usually means the order of $a$ in the multiplicative group of non-zero divisors in $\mathbb{Z}/n\mathbb{Z}$.

By Lagrange's Theorem, $\text{ord}(x) \mid \text{ord}(G)$, when $G$ is finite. In a number theoretic context, this proves Fermat's Little Theorem and Euler's generalization.

This article is a stub. Help us out by expanding it.

See also