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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.

Damien R. Sullivan