# Thread: Quotient Groups - Dummit and Foote, Section 3.1, Exercise 17

1. ## Quotient Groups - Dummit and Foote, Section 3.1, Exercise 17

I am reading Dummit and Foote Section 3.1: Quotient Groups and Homomorphisms.

Exercise 17 in Section 3.1 (page 87) reads as follows:

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Let G be the dihedral group od order 16.

$\displaystyle G = < r,s \ | \ r^8 = s^2 = 1, rs = sr^{-1} >$

and let $\displaystyle \overline{G} = G/<r^4>$ be the quotient of $\displaystyle G$ generated by $\displaystyle r^4$.

(a) Show that the order of $\displaystyle \overline{G}$ is 8

(b) Exhibit each element of $\displaystyle \overline{G}$ in the form $\displaystyle \overline{s}^a \overline{r}^b$

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I have a problem with part (b) in terms of how you express each element of $\displaystyle \overline{G}$ in the form requested - indeed, I am not quite sure what is meant by "in the form $\displaystyle \overline{s}^a \overline{r}^b$"

My working of the basics of the problem was to put $\displaystyle H = <r^4>$ and generate the cosets of H as follows:

$\displaystyle 1H = H = \{ r^4, 1 \}$

$\displaystyle rH = \{ r^5, r \}$

$\displaystyle r^2H = \{ r^6, r^2 \}$

$\displaystyle r^3H = \{ r^7, r^3 \}$

$\displaystyle sH = \{ sr^4, s \}$

$\displaystyle srH = \{ sr^5, sr \}$

$\displaystyle sr^2H = \{ sr^6, sr^2 \}$

$\displaystyle sr^3H = \{ sr^7, sr^3 \}$

So the order of $\displaystyle \overline{G}$ is 8

BUT - how do we express the above in the form $\displaystyle \overline{s}^a \overline{r}^b$ and what does the form mean anyway?

Would appreciate some help.

Peter

2. ## Re: Quotient Groups - Dummit and Foote, Section 3.1, Exercise 17

By representing each of the elements in the form $\displaystyle \overline{s}^a\overline{r}^b$, they simply mean choosing a coset representative. One element from each distinct coset.

I.e., The coset $\displaystyle sr^3H = \{sr^7,sr^3 }$ would be represented by either $\displaystyle \overline{s}\overline{r}^3$ or $\displaystyle \overline{s}\overline{r}^7$