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State/Lagranges theorem
From SlugmathWiki
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 $\vert G \vert$.
Logical Connections
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
The following statements and definitions rely on the material of this page: Def/Order (permutation), and State/Groups of prime order are cyclic
To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/Basic group theory
