Difference between revisions of "Order (group theory)"
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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 <math>\text{ord(G)}</math>, 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>. | ||
Latest revision as of 09:43, 11 April 2020
In group theory, the term order has different meanings in different contexts.
The order of a group 
, sometimes denoted 
, is the cardinality of its underlying set.
The order of an element 
of , 
, is the order of the subset generated by . If is finite, then it is also the least positive integer 
for which 
.
In number theory, for 
relatively prime to , the order of (mod ) usually means the order of in the multiplicative group of non-zero divisors in 
.
By Lagrange's Theorem, 
, when is finite. In a number theoretic context, this proves Fermat's Little Theorem and Euler's generalization.
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