In learning a new category it helps to have examples of things not in it, as
well as things that are. So what are things which might look like groups but
aren't, and why? As we've already touched upon, *all* the fractions or
reals under multiplication aren't a group, because zero has no inverse. The
integers under multiplication aren't either, for the different reason that no
integer except 1 and -1 has an inverse under multiplication. The negative
numbers under multiplication aren't; no identity or inverse, plus it's not
closed. The non-negative numbers under addition lack inverses; the positive
numbers under addition also lack an identity. The integers under division
lack closure (or, depending on how you look at it, a fully defined operation.)
Numbers under absolute value aren't; absolute value isn't even a binary
operation.