Is $(Q,.)$ an abelian and cyclic group?

Question

My teacher said that an example of an abelian group that is not cyclic is $(Q/\{0\},.)$.

So I was curious why did he use $Q/\{0\}$ instead of just $Q$. Does using $Q$ make it cyclic?

Answer 1

As pointed out, $\Bbb Q \color{red}{\setminus} \{0\}$ was used since $(\Bbb Q, \cdot)$ is not a group because $0$ has no multiplicative inverse.

$(\Bbb Q \color{red}{\setminus} \{0\}, \cdot)$ is indeed abelian since the usual multiplication of rationals is commutative. However, it is not cyclic.
To see this, suppose that $r \in \Bbb Q \color{red}{\setminus} \{0\}$ was a generator. Then, $r^n = 2$ and $r^m = 3$ for some integers $n$ and $m$.
By considering $2^m$ and $3^n$, arrive at a contradiction.

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Note the slash that I have used. $\Bbb Q \color{red}{\setminus} \{0\}$ denotes the set of non-zero rationals. The slash which you have used has a different meaning.

Answer 2

No.

Every group has exactly one idempotent, namely its identity. For suppose $x^2=x$ is an idempotent. Then $xx=x^2=x=ex$, so multiplying on the right by $x^{-1}$ gives $x=e$.

In $(\Bbb Q,\cdot)$ both $0\cdot 0=0$ and $1\cdot 1=1$ but $0\neq 1$. Hence $(\Bbb Q,\cdot)$ is not a group.

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An example of an abelian group that is not cyclic is the Klein four group.