Order (group theory)

In group theory, the term order has different meanings in different contexts.

The order of a group $G$ , sometimes denoted $\text{ord(G)}$ , 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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Order (group theory)

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