The Group Z/2Z x Z/2Z is not Cyclic

The Mathematician — Fri, 25 Nov 2022 13:00:47 +0000

$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ is not cyclic

Proof. We will explicitly determine if the group

\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}

can be generated by one of its own elements. Recall that we have the group

(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, +)

and that

\begin{align*} \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} = \{(0,0),(1,0),(0,1),(1,1)\}. \end{align*}

Let’s check if

(1,0)

is the generator of

\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}

. Then

\begin{align*} (1,0) + (1,0) = (0,0), \quad (0,0) + (1,0) = (1,0). \end{align*}

So we see that

\langle (1,0) \rangle = \{(0,0),(1,0)\}

which is not the generator of

\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}

. Now we can check all the other elements:

\begin{align*} \langle (0,0) \rangle &= \{(0,0)\} \\ \langle (0,1) \rangle &= \{(0,0),(0,1)\} \\ \langle (1,1) \rangle &= \{(0,0),(1,1)\} \end{align*}

No single element of the group

\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}

can generate the group

\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}

, so

\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}

is not cyclic.

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