"A Book of Abstract Algebra" presents this exercise:

In each of the following, $H$ is a subgroup of $G$. List the cosets of $H$. For each coset, list the elements of the coset.

$G=\mathbb{Z}_{15}, H=\langle 5 \rangle $

My attempt follows to calculate the Right and Left Cosets:

$$H + 5 = \langle 10 \rangle $$

Is this correct? If not, please let me know how to figure out the cosets of $H$ in this problem.

  • $\begingroup$ This is not a simple here's an equation, here's the answer involved. You have to consider $g+H$ for all $g \in G$ (so you gotta look at $1+H$, $2+H$, etc.). $\endgroup$ – Clarinetist Jul 13 '15 at 3:03

The cosets are going to be $1+H,2+H,3+H,4+H,H$.

Notice that $H=\left<5\right>={0,5,10}$, so the cosets are: $$\{0,5,10\}\\\{1,6,11\}\\\{2,7,12\}\\\{3,8,13\}\\\{4,9,14\}$$ Because $x+H=\{x+h:h\in H\}$

EDIT: Also, you need not worry about $6+H$ et al because the higher elements cycle back down nicely.

  • $\begingroup$ What's the difference between $\mathbb{Z}_5$ and $<5>$? Is the latter simply a notation once $\mathbb{Z}$ has been introduced? $\endgroup$ – Kevin Meredith Jul 19 '15 at 21:22
  • 1
    $\begingroup$ @KevinMeredith: $\langle 5 \rangle$ refers to the subgroup generated by repeatedly adding 5 and its inverses to itself. So in $\mathbb{Z}_{15}$, $\langle 5 \rangle = \{0, 5, 10\}$ and is a subgroup with the inherited operation. On the other hand, if you consider $\langle 5 \rangle$ as a subgroup of $\mathbb{Z}$, this is simply all positive and negative multiples of 5. $\endgroup$ – jackson h Jul 19 '15 at 22:39

The left cosets of $H$ are $1+H$, $2+H$, $3+H$, $4+H$ and $H$ itself. Because $\mathbb{Z}_{15}$ is Abelian, these are the same as the right cosets.

As for what's in the cosets... $$x+H = \{x+h:h \in H\}$$


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