# Thread: abstract algebra - dihedral groups

1. ## abstract algebra - dihedral groups

Can someone help me with this proof? It was an exam problem, but I can't get it.

D_2n = < r, s | r^n = s^2 = 1, rs = sr^(-1) >
If n = 2k is even and n>= 4, show that z = r^k is an element of order 2 which commutes with all elements of D_2n. Show also that z is the only nonidentity element of D_2n which commutes with all elemtns of D_2n.

2. Originally Posted by dori1123
Can someone help me with this proof? It was an exam problem, but I can't get it.

D_2n = < r, s | r^n = s^2 = 1, rs = sr^(-1) >
If n = 2k is even and n>= 4, show that z = r^k is an element of order 2 which commutes with all elements of D_2n. Show also that z is the only nonidentity element of D_2n which commutes with all elemtns of D_2n.
Hint: Use the fact that $\displaystyle r^n s = r^{-n} s$

3. For the first proof, should I show that r^k is an element of order 2 or should I show that r^k commutes with all elements of D_2n?

Can you help me to get started? Thank you.

4. Originally Posted by dori1123
For the first proof, should I show that r^k is an element of order 2 or should I show that r^k commutes with all elements of D_2n?
Note that $\displaystyle r^k \not = e$ -- why?
Now $\displaystyle (r^k)^2 = r^{2k} = e$.
Thus the order of $\displaystyle r^k$ is two.
This means that $\displaystyle (r^k)(r^k)^{-1} = e$.
Thus, $\displaystyle r^{-k} = r^k$.

Any element in the dihedral group has form $\displaystyle r^as^b$.
Now $\displaystyle (r^as^b)r^k = r^a(s^br^k) = r^a(r^{-k}s^b) = r^a(r^ks^b) = r^k(r^as^b)$.
Thus $\displaystyle r^k$ commutes with everything.

5. Originally Posted by ThePerfectHacker
Note that $\displaystyle r^k \not = e$ -- why?
Now $\displaystyle (r^k)^2 = r^{2k} = e$.
Thus the order of $\displaystyle r^k$ is two.
So I know that $\displaystyle r^k \not = e$ and $\displaystyle (r^k)^2 = r^{2k} = e$ by the definition of $\displaystyle D_{2n}$, because $\displaystyle n = 2k$, so $\displaystyle r^n = r^{2k} = (r^k)^2 = e$?

6. Originally Posted by dori1123
So I know that $\displaystyle r^k \not = e$ and $\displaystyle (r^k)^2 = r^{2k} = e$ by the definition of $\displaystyle D_{2n}$, because $\displaystyle n = 2k$, so $\displaystyle r^n = r^{2k} = (r^k)^2 = e$?
Yes

7. How do I show that z is the only nonidentity element of $\displaystyle D_{2n}$ which commutes with all elements of $\displaystyle D_{2n}$?

8. Originally Posted by dori1123
How do I show that z is the only nonidentity element of $\displaystyle D_{2n}$ which commutes with all elements of $\displaystyle D_{2n}$?
Hint: The element which commutes with all element if and only if it commutes with $\displaystyle r,s$. Thus find which element commutes with $\displaystyle r,s$.

9. Obviously, the identity element commutes with all elements in $\displaystyle D_{2n}$. But I am looking for a nonidentity element, so from the first proof, $\displaystyle r^k$ is the nonidentity element that commutes with all elements in $\displaystyle D_{2n}$. But how do I know that it is the ONLY nonidentity that commutes with those elements?

10. Originally Posted by dori1123
But how do I know that it is the ONLY nonidentity that commutes with those elements?
Look at what I wrote above! Let $\displaystyle x$ be an element in this group then we can write it as $\displaystyle x=r^as^b$ where $\displaystyle 0\leq a\leq n-1, 0\leq b\leq 1$. Now $\displaystyle r^as^b$ commute with all the element if and only if it it commutes with the generating set i.e. it commute with $\displaystyle r$ and $\displaystyle s$. Thus, find the conditions on $\displaystyle a,b$ so that $\displaystyle r^as^b$ commute with $\displaystyle r$ and $\displaystyle s$.

11. So I let $\displaystyle x = r^as^b$ where $\displaystyle 0<=a<=n-1, 0<=b<=1$, then $\displaystyle (r^as^b)r = r^a(s^br) = r^a(r^{-1}s^b) = r^ars^b = r(r^as^b)$ and $\displaystyle (r^as^b)s = (r^as)s^b = sr^{-a}s^b = s(r^as^b)$. So $\displaystyle r^as^b$ commutes with all elements of $\displaystyle D_{2n}$ no matter what a and b are, isn't it? Then the nonidentity element in the form of $\displaystyle r^as^b$ also commutes with all elements of $\displaystyle D_{2n}$, so $\displaystyle r^a$ is not the only nonidentity element that commutes with the elements in $\displaystyle D_{2n}$...?
Am I thinking this right?...

12. Originally Posted by dori1123
So I let $\displaystyle x = r^as^b$ where $\displaystyle 0<=a<=n-1, 0<=b<=1$, then $\displaystyle (r^as^b)r = r^a(s^br) = r^a(r^{-1}s^b) = r^ars^b = r(r^as^b)$ and $\displaystyle (r^as^b)s = (r^as)s^b = sr^{-a}s^b = s(r^as^b)$. So $\displaystyle r^as^b$ commutes with all elements of $\displaystyle D_{2n}$ no matter what a and b are, isn't it? Then the nonidentity element in the form of $\displaystyle r^as^b$ also commutes with all elements of $\displaystyle D_{2n}$, so $\displaystyle r^a$ is not the only nonidentity element that commutes with the elements in $\displaystyle D_{2n}$...?
Am I thinking this right?...
Let us consider two cases: (i)$\displaystyle x=r^a$, $\displaystyle 0\leq a\leq n-1$ (ii)$\displaystyle x=r^as$, $\displaystyle 0\leq s\leq 1$.

In case (i) we have that $\displaystyle x=r^a$ commutes with $\displaystyle s,r$. It definitely commutes with $\displaystyle r$ no matter what $\displaystyle a$ is but to commute with $\displaystyle s$ we need to have $\displaystyle r^a s = sr^a = r^{-a}s$ and so $\displaystyle a\equiv -a ~ (n)$ this forces $\displaystyle a = k$ (where $\displaystyle k$ is such an integer so $\displaystyle 2k=n$). Thus, we find that $\displaystyle x=r^k$ commutes with all the elements.

You try case (ii).

13. So, $\displaystyle (r^as)s = (sr^{-a})s = (sr^a)s = s(r^as)$. So $\displaystyle r^as$ commutes with s. And $\displaystyle (r^as)r = (sr^{-a})r = (sr^a)r$. Here is what I wonder, if I am given $\displaystyle r^as = sr^{-a}$, does that mean $\displaystyle sr^a = r^{-a}s$?

14. Originally Posted by dori1123
So, $\displaystyle (r^as)s = (sr^{-a})s = (sr^a)s = s(r^as)$. So $\displaystyle r^as$ commutes with s. And $\displaystyle (r^as)r = (sr^{-a})r = (sr^a)r$. Here is what I wonder, if I am given $\displaystyle r^as = sr^{-a}$, does that mean $\displaystyle sr^a = r^{-a}s$?
$\displaystyle (r^a s)s = r^a s^2 = r^a$.
$\displaystyle s(r^a s) = (sr^a)s = (r^{-a} s)s = r^{-a}$.
This means if we want commutavity then $\displaystyle r^a = r^{-a}$ i.e. $\displaystyle a=0,k$.

Now there are two cases: (i)$\displaystyle x=r^0 s = s$ (ii)$\displaystyle x=r^k s$.
Which of these commute with the remaining element $\displaystyle r$?

15. Originally Posted by ThePerfectHacker
Now there are two cases: (i)$\displaystyle x=r^0 s = s$ (ii)$\displaystyle x=r^k s$.
Which of these commute with the remaining element $\displaystyle r$?
So for $\displaystyle x = r^0s = s$, $\displaystyle xr = sr$ does not equal to $\displaystyle rx = rs = sr^{-1}$ (unless $\displaystyle r = r^{-1}$?). For $\displaystyle x = r^ks$, $\displaystyle xr = r^ksr = sr^{-k}r = sr^{-k+1}$ and $\displaystyle rx = rr^ks = r^{k+1}s$, so $\displaystyle r^ksr$ does not equal to $\displaystyle rr^ks$. Right?

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