Definition:Order of Group Element/Definition 2
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Definition
Let $G$ be a group whose identity is $e_G$.
Let $x \in G$ be an element of $G$.
The order of $x$ (in $G$), denoted $\order x$, is the order of the group generated by $x$:
- $\order x := \order {\gen x}$
Also known as
Some sources refer to the order of an element of a group as its period.
Also denoted as
The order of an element $x$ in a group is sometimes seen as $\map o x$.
Some sources render it as $\map {\operatorname {Ord} } x$.
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Subgroups: Example $28$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 41$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups