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 10:43, 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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