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Describe the cosets of the subgroup $\langle 3\rangle$ of $\mathbb{Z}$

The problem I have is $\mathbb{Z}$ is infinite.

So we know that $\langle 3\rangle=\{0,3,6,9,12,\ldots\}$ and I know the definition of cosets (in this case right cosets) is the set of all products of ha, as a remains fixed and $h$ ranges over $H$.

So $H$ is the subgroup $\langle 3\rangle$ which does not remain fixed and the elements of $\mathbb{Z}$ remains fixed.

I started writing some numbers out to see how I can possibly describe it but I didn't see any help.

I defined: $\mathbb{Z}=\{\ldots,-4,-3,-2,-1,0,1,2,3,4,\ldots\}$ and I already defined $\langle 3\rangle=\{0,3,6,9,12,\ldots\}$

Therefore here are some numbers:

Lets pick $-4$ as our fixed $a$, so then we obtain:

$0 \times -4=0$

$3 \times -4=-12$

$6 \times -4=-24$

and so on

If I pick $2$ as our fixed $a$, then we have:

$0 \times 2=0$

$3 \times 2=6 $

$6 \times 2=12 $

I probably can't see it but then how would I describe the cosets?

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    $\begingroup$ I change $<3>$ to $\langle 4\rangle$ and in other ways civilized the typesetting in this posting. ${}\qquad{}$ $\endgroup$
    – Michael Hardy
    Commented Apr 1, 2015 at 4:30

2 Answers 2

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The operation on $Z$ in this case is ADDITION. So $$ <3> = \{ \ldots, -6, -3, 0, 3, 6, \ldots \} $$ and a typical coset is $$ <3> + 0 = <3> $$ while another is $$ <3> + 1 = \{ \ldots, -6+1, -3+1, 0+1, 3+1, 6+1, \ldots\}. $$ With that, can you write down all the other cosets? How many are there?

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  • $\begingroup$ I dont know why my mind was thinking multiplication but thanks for pointing it out! Well, if I write down all the other cosets wouldn't that be sheer madness since I probably will use all the paper I own! With that being said, there are infinite cosets? In the form <3>+n={...-6+n,-3+n,0+n,3+n,6+n...} where n is $\in \mathbb{Z}$ $\endgroup$
    – Justin
    Commented Apr 1, 2015 at 19:41
  • $\begingroup$ Well, if you wrote down the coset for $n = 4$, it'd contain $\{ \ldots, -6+4, -3+4, 0+4, 3+4, \ldots \} = \{ \ldots, -2, 1, 4, 7, \ldots \}$. which is exactly the same as the coset for $n = 1$. If you wrote down $<3> + n$ for every $n$, it'd take infinitely long,. But you'd write each coset many many times. To write an exhaustive list of cosets -- not of coset descriptions -- should take only a moment or two. $\endgroup$
    – John Hughes
    Commented Apr 1, 2015 at 20:22
  • $\begingroup$ I see the pattern now! $\endgroup$
    – Justin
    Commented Apr 2, 2015 at 2:28
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Since $\Bbb Z$ is an additive, your cosets are $a + \Bbb \langle 3 \rangle$, not $a\langle 3 \rangle$, like you're calculating.

So, to given an example, one would be the coset $1 + \langle 3 \rangle = \{1 + n: n \in \langle 3 \rangle\} = \{\ldots, -5, -2, 1, 4, \ldots\}$.

So that was your only problem, not using the right group operation; everything else looked good!

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  • $\begingroup$ Got it! Thanks for everything! $\endgroup$
    – Justin
    Commented Apr 1, 2015 at 19:42

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