Why are the nonzero integers under multiplication NOT a group? [closed]

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I was just checking out a GRE mathematics practise test. I have never had theories about groups, so could someone explain how to test if something is a group and why the 'nonzero integers under multiplication' is not a group while the following are:

1. The integers under addition
2. The nonzero real numbers under multiplication
3. The complex numbers under addition
4. The nonzero complex numbers under multiplication

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2 Answers

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1. A group consists of a set of things that can be combined using a particular operation.

2. The set of things must contain a neutral element. For addition, the neutral element is 0. For multiplication, the neutral element is 1. The neutral element an element that if combined with any other element $x$, produces $x$ again.

3. Every element in the set must have an inverse. If you combine any element $x$ with its inverse, you get the neutral element. For addition, the inverse is given by negation since $x + (-x) = 0$. For multiplication, the inverse is given by reciprocal, since $x \cdot (\frac{1}{x}) = 1$.

"Nonzero integers under multiplication" are not a group. They fit criteria (1) and (2). In fact, the neutral element is 1. But they don't fit (3) because there are no inverses. For example, you can't find any nonzero integer you can multiply with 7 in order to get 1 back. This means that 7 has no inverse.

Because not all elements have inverses, this set doesn't constitute a group.

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The reason non-zero integers under multiplication are not a group is because there is no inverse element for all $2, 3, \ldots .$

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