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- Jun 22, 2012

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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.

[TEX] G = < r,s \ | \ r^8 = s^2 = 1, rs = sr^{-1} > [/TEX]

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

(a) Show that the order of [TEX] \overline{G} [/TEX] is 8

(b) Exhibit each element of [TEX] \overline{G} [/TEX] in the form [TEX] \overline{s}^a \overline{r}^b [/TEX]

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

My working of the basics of the problem was to put [TEX] H = <r^4> [/TEX] and generate the cosets of H as follows:

[TEX] 1H = H = \{ r^4, 1 \} [/TEX]

[TEX]rH = \{ r^5, r \}[/TEX]

[TEX]r^2H = \{ r^6, r^2 \}[/TEX]

[TEX]r^3H = \{ r^7, r^3 \}[/TEX]

[TEX]sH = \{ sr^4, s \}[/TEX]

[TEX]srH = \{ sr^5, sr \}[/TEX]

[TEX]sr^2H = \{ sr^6, sr^2 \}[/TEX]

[TEX] sr^3H = \{ sr^7, sr^3 \} [/TEX]

So the order of [TEX] \overline{G} [/TEX] is 8

Would appreciate some help.

Peter

[Note: This has also been posted on MHF]

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

-------------------------------------------------------------------------------------------------------------

Let G be the dihedral group od order 16.

[TEX] G = < r,s \ | \ r^8 = s^2 = 1, rs = sr^{-1} > [/TEX]

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

(a) Show that the order of [TEX] \overline{G} [/TEX] is 8

(b) Exhibit each element of [TEX] \overline{G} [/TEX] in the form [TEX] \overline{s}^a \overline{r}^b [/TEX]

------------------------------------------------------------------------------------------------------------------

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

My working of the basics of the problem was to put [TEX] H = <r^4> [/TEX] and generate the cosets of H as follows:

[TEX] 1H = H = \{ r^4, 1 \} [/TEX]

[TEX]rH = \{ r^5, r \}[/TEX]

[TEX]r^2H = \{ r^6, r^2 \}[/TEX]

[TEX]r^3H = \{ r^7, r^3 \}[/TEX]

[TEX]sH = \{ sr^4, s \}[/TEX]

[TEX]srH = \{ sr^5, sr \}[/TEX]

[TEX]sr^2H = \{ sr^6, sr^2 \}[/TEX]

[TEX] sr^3H = \{ sr^7, sr^3 \} [/TEX]

So the order of [TEX] \overline{G} [/TEX] is 8

*- how do we express the above in the form [TEX] \overline{s}^a \overline{r}^b [/TEX] and what does the form mean anyway?***BUT**Would appreciate some help.

Peter

[Note: This has also been posted on MHF]

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