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What it says

Order of Element Divides Group Order

For any finite group $G$ and any element $x$ in $G$, the order of $x$ divides the cardinality of $G$, i.e., $\operatorname{ord}(x) \mid \#G$.

What it says

Lagrange's Theorem: Element Order Divides Group Order

For any element $x$ in a group $G$, the order of $x$ divides the cardinality of $G$, i.e., $\operatorname{orderOf}(x) \mid \operatorname{Nat.card}(G)$.

What it says

Order of Element Divides Group Cardinality in Finite Additive Groups

For any element $x$ in a finite additive group $G$, the additive order of $x$ divides the cardinality of $G$, i.e., $\text{addOrderOf}(x) \mid \#G$.

What it says

Order of Element Divides Group Cardinality in Additive Groups

For any element $x$ in a finite additive group $G$, the order of $x$ divides the cardinality of $G$. In symbols, $\operatorname{addOrderOf}(x) \mid \#G$, where $\#G$ denotes the cardinality of $G$ as a natural number.

What it says

Order of Element Divides Subgroup Size: $ \text{orderOf}(x) \mid |s| $

For any group $G$, subgroup $s \subseteq G$, and element $x \in s$, the order of $x$ divides the cardinality of $s$, i.e., $\text{orderOf}(x) \mid |s|$.