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Math Help - Quotient Groups - Dummit and Foote, Section 3.1, Exercise 17

  1. #1
    Super Member Bernhard's Avatar
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    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:

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

    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 8


    BUT - 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
    Last edited by Bernhard; November 20th 2013 at 12:04 AM.
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  2. #2
    Member Haven's Avatar
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    Re: Quotient Groups - Dummit and Foote, Section 3.1, Exercise 17

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

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