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

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

Would appreciate some help.

Peter