Abstract Algebra – Why (R*, ×) is a Group but (R, ×) is Not

Question

$\mathbb R^*$ refers to $\mathbb R$ without zero.

Please explain the statement "$\mathbb R$ is not a group under multiplication, it is a group under addition." in the comment. Basically:

1. Why is multiplicative group $(\mathbb R^*, ×)$ a group?

2. Why is $(\mathbb R, ×)$ not group?

3. Why are $(\mathbb R^*, +)$ and $(\mathbb R, +)$ groups?

Answer 1

$(\mathbb{R}^*, \times)$ is a group, since

• It has a neutral element $1$
• It has inverses $1\over x$ for all $x \in \mathbb{R}^*$
• The operation is associative

$(\mathbb{R}, \times)$ is not a group, because $0$ has no multiplicative inverse.

$(\mathbb{R}, +)$ is a group, since

• It has a neutral element $0$
• It has inverses $-x$ for all $x \in \mathbb{R}$
• The operation is associative

$(\mathbb{R}^*, +)$ is not a group, since it is not closed under operation: $x+(-x)=0 \not \in \mathbb{R}^*$