Let $H=\{[0]_8, [4]_8\} \subset \Bbb Z_8$ and $[a]_8 \sim [b]_8$ if $-[a]_8 +[b]_8 \in H$. Determine the elements in the equivalence class of $[1]_8$.

Question

Let $H=\{[0]_8, [4]_8\}$ be a subgroup of $\Bbb Z_8$ and let $[a]_8 \sim [b]_8$ if $-[a]_8 +[b]_8 \in H$. Determine the elements in the equivalence class of $[1]_8$.

I’m slightly confused about how I can find the elements of the equivalence class of $[1]_8$. The elements of $[1]_8$ are the cosets $1+8\Bbb Z$ so $[1]_8 = \{..., -9,...,1,...9,... \}$ so am I supposed to use these elements to determine the equivalence classes?

Answer 1

Hint: It helps to sound out what it is saying: "The elements of the $\sim$-equivalence class of $[1]_8$ are precisely those elements of $\Bbb Z_8$ that differ from $[1]_8$ by an element of $H$."

The equivalence class is $$\{[1]_8, [5]_8\}.$$