Is the set a group wrt multiplication. If not, what all conditions does it fail? If so, what's the order.

The set {[1], [3]} ⊆ Z(sub)8.

The book says it's a group of order 2.

I made a Cayley table but have no clue how to check associativity, and inverse. I know it's closed by theorem.

But how do I check associativity? I'm confused.

If {[1]} = {[1], [2], [3], [4], [5], [6], [7]}, and

{[3]} = {[3], [6], [1], [4], [7], [2], [5]}, then is

{[1], [3]} = [1]}? I don't know the relevance of this. It's just something that I am confused about.

Identity is [1], that was clear in the table.

Inverse: Rule [a] [b] = [1]. How do I check this?

Thanks for all your help. This is blowing my mind!