Cyclic group generated by empty set.

Question

I have come across a problem in Durbin, Modern Algebra, that I am not quite convinced of the answer.
The question states:

If $\emptyset$ denotes the empty set, what is $\langle\emptyset\rangle$ (in any group $G$)?

I searched for answers to this question for a while and the only one I could find stated:

$\langle\emptyset\rangle = \{e\}$ , the identity element.

I am not quite sure of the logic behind this answer and any hints or explanations would be thoroughly appreciated.

Answer 1

By definition, if $S\subset G$, $\langle S\rangle$ is the smallest subgroup of $G$ which contains $S$. Since the smallest of all subgroups of $G$ is $\{e\}$ and since $\{e\}\supset\emptyset$, $\langle\emptyset\rangle=\{e\}$.