Let be a non-zero element (i.e. ). Now consider . None of these are equal to eachother, because otherwise since it means which is impossible since we are assuming that . That means by pigeonhole principle that this set has to be a permutation of in some order. And so, there exists such that thus this means is an inverse for .

I did the hard part in (a). You should be able to do this now.(b) Deduce that , the set of non-zero elements of , is a group under multiplication mod .

Just show there exists an element that(c) Show that if is not prime, then is not a multiplicative group.does nothave an inverse. Hint: if is not prime then there exists such that , and argue that no such can be found such that .

Yes.(d) Is cyclic?