Is the empty set a group?

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.

As BrianO says, $\emptyset$ is not a group, because every group has an identity element. This also means that $\emptyset$ is not a vector space, it's not a ring, it's not a module, and it's not a boolean algebra. However, $\emptyset$ is a perfectly good: semilattice, band, and affine space. Also, it's best to drop the non-emptiness condition from the usual definition of a heap, in which case $\emptyset$ is a perfectly good heap.