# quotient group inheritance of some properties from original group

Let us say that there is the group with addition modulo 6:

$G = \{0,1,2,3,4,5\}$ and let $N = \{0,3 \}$.

Then the quotient group $G/N$ would be $G/N = \{ \{0,3\}, \{1,4 \}, \{2,5\} \}$.

According to my sources, they say that as $N$ is normal subgroup, $G/N$ must be also group. As the group $G$ is abelian, according to my sources (Wikipedia and textbook), they say that $G/N$ must be abelian group.

The question is, by quotient group $G/N$ being group, is group operation done by assuming that in the example, $0$ and $3$ are treated same, $1$ and $4$ are treated same and $2$ and $5$ are treated same? And abelian-ness can be checked by adding $(3+4) \pmod 6 = (4+3) \pmod 6 = 1 = 4?$

Or is this something else?

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Yeah, essentially you are right that you have to treat them the same. So, for example, using the fact that $1$ and $4$ are the same we have $1+2=3\text{ mod }6$ while $4+2=0\text{ mod }6$ and this makes sense as $0$ and $3$ are treated the same. You should realise though that what is really going on is that you are working $\text{mod }3$. –  user1729 Mar 28 '13 at 10:14
You may find interesting this question about quotient sets. –  A.P. Mar 28 '13 at 12:48

$$\{1,4\}+\{2,5\}=\{0,3\}\iff \bar 1+\bar 2=\bar 0$$
Yes, but the operation is not more modulo $\,6\,$ , as my example shows, and this is what the OP asked, among other things. –  DonAntonio Mar 28 '13 at 12:47