# What is the group $G=\mathbb Z_2+\mathbb Z_2$?

I'm trying to figure out what is the group $G=\mathbb Z_2+\mathbb Z_2$ where $\mathbb Z_2 = \{0,1\}$ with the operation addition modulo $2$.

I tried to find this group by adding elements from $\mathbb Z_2$ and $\mathbb Z_2$ and I got that :

$G=\mathbb Z_2+\mathbb Z_2 = \mathbb Z_2$ (because $0+0 =0, 0+1=1,1+0=1,1+1=0)$

Is it correct?

• Presumably this $\mathbb{Z}_2 \times \mathbb{Z}_2$ or (in this case, equivalently) $\mathbb{Z}_2 \oplus \mathbb{Z}_2$? – Travis Jan 17 '15 at 12:50
• @Travis What I wrote isn't logical? You can't add two groups together? – SyndicatorBBB Jan 17 '15 at 12:53
• Well, what you wrote is logical. However, you can't add two different groups together in this sense, unless they are both subgroups of a third group. Nevertheless, presumably this $\Bbb Z_2+\Bbb Z_2$ is meant as $\Bbb Z_2\oplus\Bbb Z_2$, which is the $4$ element Klein group. Can we get more information about the context of this exercise?? – Berci Jan 17 '15 at 12:55
• @Berci Can you please show me how the elements of groups $G$ look like and explain what is the difference between $\mathbb Z_2 \times \mathbb Z_2$ and $\mathbb Z_2 \oplus \mathbb Z_2$? – SyndicatorBBB Jan 17 '15 at 12:59
• @Berci You can add up two subsets of the same (additive) group, and ${\mathbb Z}_2$ is certainly a subset of ${\mathbb Z}_2$, so (with that interpretation) ${\mathbb Z}_2 + {\mathbb Z}_2 = {\mathbb Z}_2$ is correct. – Derek Holt Jan 17 '15 at 13:10