Why is this true? I don't understand.

Example 2 p164

It states that in the subgroup H={[2], [4], [6], [8]},

[8]=[2]ˉ¹ in Z₁₀ under multiplication.

This is what I gather:

Since [2] ∈ H, [8]=[2]=[2]ˉ¹.

or

Since [8] can be written as [8]³=[12]=[2], then [2]ˉ¹ (the inverse of [2]) also equals [8].

Please let me know if this is right and what definition or theorems support this. I think I heard something like this in class but I can't remember.