how do I find all Elements of a Group?

Question

I am given a Group $\mathbb{Z_{11}^*}$. a multiplicative group.

How do i find all elements of this Group?

Answer 1

The elements of $\mathbb Z_{n}^*$ are the invertible elements of the ring $(\mathbb Z_{n},+,\times)$ and it's simple to show that $$\overline k\in \mathbb Z_{n}^*\iff \gcd(k,n)=1$$

In your example $11$ is a prime hence the ring $(\mathbb Z_{11},+,\times)$ is a field so all its elements but $0$ are invertible.