Two cosets are either identical or disjoint

Dependencies:

1. Coset
2. gH = H iff g in H

Two cosets are either identical or disjoint.

Proof

\begin{align} & g_1H \cap g_2H \neq \phi \\ &\Rightarrow \exists h_1 \in H, \exists h_2 \in H, g_1h_1 = g_2h_2 \\ &\Rightarrow \exists h_1 \in H, \exists h_2 \in H, g_2^{-1}g_1 = h_2h_1^{-1} \\ &\Rightarrow g_2^{-1}g_1 \in H \\ &\Rightarrow g_2^{-1}g_1H = H \\ &\Rightarrow g_1H = g_2H \end{align}

Dependency for:

1. Lagrange's Theorem

Info:

• Depth: 3
• Number of transitive dependencies: 4

Transitive dependencies:

1. Group
2. Coset
3. Inverse of a group element is unique
4. gH = H iff g in H