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

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    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.

    $\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 $


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

    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
    Last edited by Bernhard; Nov 19th 2013 at 11:04 PM.
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    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 $\displaystyle \overline{s}^a\overline{r}^b$, 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 $\displaystyle \overline{s}\overline{r}^7$
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