Group Direct Product of Cyclic Groups/Examples/C2 x C2

Examples of Use of Group Direct Product of Cyclic Groups

Let $C_2$ denote the cyclic group of order $2$.

The group direct product $C_2 \times C_2$ is not a cyclic group.

Proof

For the group direct product of the cyclic groups $C_m$ and $C_n$ to be cyclic, it is necessary and sufficient for $m$ and $n$ to be coprime.

That is not the case here.

It is seen from Group Direct Product of $C_2 \times C_2$ that this is an instance of the Klein $4$-group.

This is not a cyclic group.

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $13$: Direct products: Remark $1$