Order (group theory)
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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