Order (group theory)

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. ! Undefined control sequence.)

, 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.

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See also

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