What is the difference between: $G= \langle \{g\}\rangle$ and $G = \langle g\rangle$?

Question

I have seen these two different notations for a cyclic group $G$ generated by an element $g\in G$. I am curious if there is a difference between the two notations mentioned above.

Answer 1

They are the same.

Let $S$ be a finite set, write $S=\{a_1,\dots,a_n\}$.
Then $\langle S\rangle=\langle\{a_1,\dots,a_n\}\rangle$ can be written as $\langle a_1,\dots,a_n \rangle$.
Hence your question applies where $S$ is a singleton set.