TITLE: Is the empty set a group?
QUESTION [8 upvotes]: I know that a set G with a binary operation $*$ is a group, if:

$a*b\in G$, for all $a, b \in G$.
$*$ is associative:

$$(a*b)*c=a*(b*c) \\ \text{for all }a, b, c\in G.$$

An identity element $e \in G$ exists, such that

$$a*e = e*a = a\\ \text{for all }a\in G.$$

For all elements $a \in G$, there exists an $a^{-1} \in G$, such that:

$$a*a^{-1} = a^{-1}*a=e.$$
Can I use that to show that the empty set is a group?

REPLY [25 votes]: So we know that universal statements are true on empty domains, and existence statements are false. Being a group requires the existence of an identity element, and since the empty set cannot satisfy this (it has no elements) it is not a group.