Groups (10)

When does multiplying give groups?

1. Multiplying positive whole numbers together won't give us a group because, although we have an identity of 1, there is no way to get back to 1 from any number, so there aren't any inverses.
   There is no whole number that you can multiply with 5, for example, that will give 1 as the answer.
2. Taking all the whole numbers gives exactly the same problem, so these don't form a group either.
3. Taking all positive whole numbers and fractions gives a system that works fine.
   Now, for example, we have 5 x 1/5 = 1 and 2/3 x 3/2 = 1. Also, the answer given by multiplying such numbers is always another positive whole number or fraction.
4. With all whole numbers and fractions, everything also works very nicely except for one thing. What is this?

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