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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