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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.
G = < r,s \ | \ r^8 = s^2 = 1, rs = sr^{-1} >
and let \overline{G} = G/<r^4> be the quotient of G generated by r^4.
(a) Show that the order of \overline{G} is 8
(b) Exhibit each element of \overline{G} in the form \overline{s}^a \overline{r}^b------------------------------------------------------------------------------------------------------------------
I have a problem with part (b) in terms of how you express each element of \overline{G} in the form requested - indeed, I am not quite sure what is meant by "in the form \overline{s}^a \overline{r}^b"My working of the basics of the problem was to put H = <r^4> and generate the cosets of H as follows:
1H = H = \{ r^4, 1 \}
rH = \{ r^5, r \}
r^2H = \{ r^6, r^2 \}
r^3H = \{ r^7, r^3 \}
sH = \{ sr^4, s \}
srH = \{ sr^5, sr \}
sr^2H = \{ sr^6, sr^2 \}
sr^3H = \{ sr^7, sr^3 \}So the order of \overline{G} is 8BUT - how do we express the above in the form \overline{s}^a \overline{r}^b and what does the form mean anyway?
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.
G = < r,s \ | \ r^8 = s^2 = 1, rs = sr^{-1} >
and let \overline{G} = G/<r^4> be the quotient of G generated by r^4.
(a) Show that the order of \overline{G} is 8
(b) Exhibit each element of \overline{G} in the form \overline{s}^a \overline{r}^b------------------------------------------------------------------------------------------------------------------
I have a problem with part (b) in terms of how you express each element of \overline{G} in the form requested - indeed, I am not quite sure what is meant by "in the form \overline{s}^a \overline{r}^b"My working of the basics of the problem was to put H = <r^4> and generate the cosets of H as follows:
1H = H = \{ r^4, 1 \}
rH = \{ r^5, r \}
r^2H = \{ r^6, r^2 \}
r^3H = \{ r^7, r^3 \}
sH = \{ sr^4, s \}
srH = \{ sr^5, sr \}
sr^2H = \{ sr^6, sr^2 \}
sr^3H = \{ sr^7, sr^3 \}So the order of \overline{G} is 8BUT - how do we express the above in the form \overline{s}^a \overline{r}^b and what does the form mean anyway?
Would appreciate some help.
Peter
[Note: This has also been posted on MHF]
Last edited: