Question

For each binary operation ∗ defined below say whether the following is

a group or not

a) Define ∗ 𝑜𝑛 𝑍 by 𝑎 ∗ 𝑏 = 𝑎 − 𝑏

b) Define ∗ 𝑜𝑛 𝑍 by 𝑎 ∗ 𝑏 = 𝑎𝑏

c) Define ∗ 𝑜𝑛 𝑅+by 𝑎 ∗ 𝑏 = 𝑎𝑏

d) Define ∗ 𝑜𝑛 𝑄 by 𝑎 ∗ 𝑏 = 𝑎𝑏

Answer 1

1. It is not a group, as this law is not associative : $a-(b-c)=a-b+c\neq (a-b)-c$ whenever $c\neq 0$,
2. It is not a group, as not every element admits an inverse in $\mathbb{Z}$ (for example for $2\in \mathbb{Z}$ there is no $n\in \mathbb{Z}$ such that $2n=1$),
3. It is a group, as the group law is associative (the multiplication in $\mathbb{R}$ is associative), there is a unity $1\in \mathbb{R}^+$ and every element admits an inverse (as there is an inverse in $\mathbb{R}$ and for $a>0$ the inverse $a^{-1}>0$, so it is also in $\mathbb{R}^+$ ),
4. This law is associative and there is a unity, however it is not a group, as the element $0\in\mathbb{Q}$ does not admit an element $p\in\mathbb{Q}$ such that $0\cdot p =1$, so it is not invertible.