Check if pair is group

Question

On $Q \setminus 0$ is defined operation with $a * b = \frac {ab} 8$. How can I check if $(Q\setminus{0},*)$ is group? Is it commutative and how to find inverse element of $5$? Also how can I prove that $S=\{3^z \vert z \in \mathbb{Z} \}$ is normal subgroup of $Q$?

Answer 1

1) For that, you need to check the group axioms. The given set is abviously closed w.r.t. this operation, also it is clearly commutative, since the usual multiplication on $\mathbb{Q}$ is. A unit $e$ should fulfill $a * e = ae/8 = a$, I think you will figure out from that what the unit is. Same thing for inverses: $5'*5 =5'\cdot 5/8=e$, with $5'$ the inverse of 5

2) To show $S$ is a subgroup, you need to show closedness under inversion and the the above group operation; that will need a few calculations. Then, note that in a commutative group, every subgroup is automatically normal (why is this?).

If there are any more problems, just ask in the comments, but I hope that you will fill out the gaps by yourself. Greetings,

Intergalakti