Lagrange's Theorem

Dependencies:

1. Coset
2. Size of coset equals size of subset
3. Two cosets are either identical or disjoint

Let $H$ be a subgroup of $G$. Then $|H| \mid |G|$ and number of left cosets of $H$ is $\frac{|G|}{|H|}$.

Proof

$\operatorname{id}(G) \in H \implies \forall g \in G, g \in gH \implies \bigcup_{g \in G} gH = G$

Therefore, the cosets of $H$ span $G$.

Since all cosets are disjoint, size of each coset is $|H|$ and cosets span $G$, there are $\frac{|G|}{|H|}$ cosets, each of size $|H|$.

This also implies $|H| \mid |G|$.

Dependency for:

1. Order of element divides order of group

Info:

• Depth: 4
• Number of transitive dependencies: 6

Transitive dependencies:

1. Group
2. Coset
3. Size of coset equals size of subset
4. Inverse of a group element is unique
5. gH = H iff g in H
6. Two cosets are either identical or disjoint