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I have just learned about cosets and meet with the following question.

Find all left cosets of the subgroup generated by $\overline a$ in $\mathbb Z_{12}$. I know $\mathbb Z_{12} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$. I would like to know (for example if $a=2$). What does $\overline a$ mean here? Thanks.

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Do you want to know how to find the cosets? Or are you just confused about the notation? –  Dylan Moreland Dec 18 '11 at 3:28
    
yes am confused with the notation –  neema Dec 18 '11 at 3:53
1  
I see. It seems like Brad's answer should clear things up, then. Just a note: later on you'll learn that $\mathbf Z_{12}$ is the (quotient) group formed by the cosets of $12\mathbf Z$ in the group $\mathbf Z$, so you're looking at cosets of cosets here! –  Dylan Moreland Dec 18 '11 at 3:54

1 Answer 1

The elements of $\mathbb{Z}_n$ are not really integers, but rather equivalence classes of integers. The equivalence class of an integer $a$ is often denoted $\bar a$. It is the set of all integers of the form $a+nk$, where $k$ is an integer. Using this notation,

$$\mathbb{Z}_n = \{\bar 0, \bar 1, \ldots, \overline{n-1}\}.$$ For example, in $\mathbb{Z}_{12}$, we have $\overline{12} = \bar 0$ and $\overline{21} = \bar 9$. However, we often abuse notation and write things like "21 = 9" in $\mathbb{Z}_{12}$ instead.

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thanks ,i need more help i do not know how to operate those numbers, may u tell me where i can learn more about that? –  neema Dec 18 '11 at 3:58
    
i fail to find them, may u help me please? –  neema Dec 18 '11 at 4:16
    
i understand u, but how to get for example 0+12? –  neema Dec 18 '11 at 4:31
    
@neema: In $\mathbb{Z}_n$, $\overline{a}+\overline{b}=\overline{a+b}$. So in $\mathbb{Z}_{12}$, $\overline{0}+\overline{12} = \overline{0+12} = \overline{12} = \overline{0} = \overline{24} = \cdots$ –  Arturo Magidin Dec 18 '11 at 4:59
    
okay, thank u very much,may God bless u all –  neema Dec 18 '11 at 5:13

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