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Thread: [SOLVED] Show that any subgroup of <a> is normal in a dihedral group D_n

  1. #1
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    [SOLVED] Show that any subgroup of <a> is normal in a dihedral group D_n

    Let the dihedral group $\displaystyle D_n$ be given by elements $\displaystyle a$ of order $\displaystyle n$ and $\displaystyle b$ of order $\displaystyle 2$, where $\displaystyle ba=a^{-1}b$. Show that any subgroup of $\displaystyle <a>$ is normal in $\displaystyle D_n$.
    I have to say I'm not doing a good job of interpreting my textbook, here, which leaves me rather lost to prove what it asks. But here's a shot...

    If I read the previous chapter correctly, a dihedral group $\displaystyle D_n=\{a^pb^q\}$, for all integers $\displaystyle p,q$ with $\displaystyle 0\leq p<n$ and $\displaystyle q\in\{0,1\}$. Hopefully this much is correct.

    In order for the subgroup $\displaystyle <a>$ to be normal, we need to show that for each element $\displaystyle g\in D_n$, it is true that $\displaystyle g^{-1}a^rg=a^s$ for any integer $\displaystyle r\in[0,n)$ and some other integer $\displaystyle s\in[0,n)$. This clearly holds for all $\displaystyle g=a^pb^0=a^p$, since $\displaystyle (a^p)^{-1}a^ra^p=a^{n-p}a^ra^p=a^r$.

    So now we need to show that $\displaystyle (a^pb)^{-1}a^r(a^pb)=a^s$. Now, we know that since $\displaystyle o(b)=2$ then $\displaystyle b^{-1}=b$. That means $\displaystyle (a^pb)^{-1}=ba^{n-p}$. So now we evaluate $\displaystyle ba^{n-p}a^ra^pb=ba^rb$.

    Note that since $\displaystyle ba=a^{-1}b=a^{n-1}b$, then $\displaystyle ba^rb=a^{n-1}ba^{r-1}b$. And we can get by repeating this:

    $\displaystyle ba^rb=a^{n-1}ba^{r-1}b$

    $\displaystyle =a^{2n-2}ba^{r-2}b$

    $\displaystyle =a^{3n-3}ba^{r-3}b$

    $\displaystyle =a^{4n-4}ba^{r-4}b$

    ...and so on, up to:

    $\displaystyle =a^{rn-r}ba^{r-r}b=a^{rn-r}\in D_n$

    Q.E.D. ... ?

    Have I done this correctly? If not, how did I err? If so, is there a quicker/easier way to do it?

    Thanks for your help, guys.
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  2. #2
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    Quote Originally Posted by hatsoff View Post
    I have to say I'm not doing a good job of interpreting my textbook, here, which leaves me rather lost to prove what it asks. But here's a shot...
    In general if $\displaystyle G$ is a finite group and $\displaystyle H$ is a subgrop with $\displaystyle (G:H) = 2$ then $\displaystyle H$ is a normal subgroup.
    Try proving this instead.
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  3. #3
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    the problem is asking you to prove that every subgroup of <a> is normal. let $\displaystyle <a^k>=H$ be any subgroup of $\displaystyle <a>$ and $\displaystyle g \in D_n.$ let $\displaystyle h=a^{kr} \in H.$ we have $\displaystyle ghg^{-1}=(gag^{-1})^{kr}.$

    let $\displaystyle g=a^ib^j.$ if $\displaystyle j=0,$ then $\displaystyle gag^{-1}=a$ and thus $\displaystyle ghg^{-1}=a^{kr}=h \in H.$ if $\displaystyle j=1,$ then $\displaystyle g=a^ib=ba^{-i},$ because $\displaystyle ab=ba^{-1}.$ therefore $\displaystyle gag^{-1}=a^{-1}$ and hence $\displaystyle ghg^{-1}=h^{-1} \in H.$
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