I have to check if the set {1, 3, 7, 9, 11, 13, 17, 19} forms a group under multiplication modulo 20, and the only idea that I've had so far is: attempt brute force on all the elements (multiplying them all out and making a massive table), followed by induction on each element, to show that powers of each of them also belong to the set.
We've only started talking about groups and mod arithmetic last week, so I'm still trying to find my way around. I'm sure there's a better way to do this, and any advice would be much appreciated.