The SlugMath Wiki is under heavy development!

# State/Lagranges theorem

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:

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