Does a group cease to exist if no members?

Yes a group needs an identity element.

what are you talking about, exactly ?

I second Carroll's answer. Another possible interpretation is facebook groups. I don't know if this is possible now, but a few years ago I had fun looking up obscure groups with no administrators, assigning myself the role of administrator, throwing out all members and then leave, so that the group disappeared.

I third Carroll's answer. On a perhaps related topic, though, I remember some students in a high school math class having trouble with 0! being equal to 1.  By way of explanation, the teacher said “how many permutations are there of zero objects? one; there it is” and he pointed with his hands into empty space.

Actually a group doesn't cease to exist if it didn't exist in the first place.