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Theorem: (Lagrange's Theorem) Suppose that $G$ is a group, and $H$ is a subgroup of $G$. Then there is an equality of cardinal numbers: $$\vert G \vert = \vert G/H \vert \cdot \vert H \vert,$$ where $G/H$ denotes the set of cosets of $H$ in $G$.
In particular, the following statements hold:
- The cardinality of $H$ divides the cardinality of $G$.
- The cardinality of $G/H$ (i.e., the index of $H$ in $G$) divides the cardinality of $G$.
When $g \in G$, and $H = \langle g \rangle$ is the subgroup generated by $g$, we find that:
- $\vert \langle g \rangle \vert$, which equals the order of $g$, divides $\vert G \vert$.
This statement logically relies on the following definitions and statements: Def/Group, Def/Subgroup, Def/Coset, Def/Cardinality, Def/Divides, State/Axiom of choice, Def/Bijection, State/Cancellation in groups
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