How many finite integer groups are there?

Question

I have got an assignment which has a question that asks us to write a program which can decide whether or not the input entered is a group or not. Now I know that the set of integers is a group on the addition operation, but that is infinite. The only finite integer group I can think of is the integers mod n. Are there any other finite groups having integer elements?

Answer 1

The only finite subgroup of $\mathbb Z$ under addition is $\{0\}$ because all other elements have infinite order.

The only subset of $\mathbb Z$ that becomes a group under multiplication is $\{0\}$, $\{1\}$ and $\{1,-1\}$.

First notice that no element other than $0,1$ and $-1$ can be present because they have infinite order.

Now notice that a subset including $0$ must only contain $0$ because otherwise the left multiplication won't be injective.

And then check the $3$ options that contain $1$ and $-1$.