Sundry Coset Results

Theorems
Let $$G$$ be a group and let $$H$$ be a subgroup of $$G$$.

Let $$x, y \in G$$.

Let:
 * $$x H$$ denote the left coset of $$H$$ by $$x$$;
 * $$H y$$ denote the right coset of $$H$$ by $$y$$.

Then the following results apply: