# Thread: what is binary dihedral group?

1. ## what is binary dihedral group?

Can anybody explain what the binary dihedral group is? Can you give an example of it?. When do you apply it?

Is Aut(C_11) a binary dihedral group? How do you express it in terms of generators and relations.

Many thanks

KN

2. Originally Posted by knguyen2005
Can anybody explain what the binary dihedral group is? Can you give an example of it?. When do you apply it?

Is Aut(C_11) a binary dihedral group? How do you express it in terms of generators and relations.

Many thanks

KN
You can look at this.
Also, $\displaystyle \text{Aut}(C_{11}) \simeq C_{10}$ and so it is not dicyclic.

3. Yes of course , because 11 is prime number.
But maybe I did not say specifically about the question. I meant when you do the semidirect product and crucial calculation.
The actual question is: describe explicitly all homomorphisms of h:C4 --> Aut(C11). This is the short answer ( I cant type all of it)
C4 ={1,y,...,y^3}
C11 = {1,x,.....,x^10}

Thus there is a unique subgroup of order 2 of C11, say t(x) = x^(-1)

So, there are 2 homomorphisms 1 and t
When doing the crucial calculation, we get 2 groups:
< X,Y :X^11 = Y^4 = 1, YX = XY> which is abelian group (1)
< X,Y : X^11 = Y^4 = 1, YXY^(-1) = X^(-1)> (2)
In the lecture, my lecturer said the second group (2) is binary dihedral group.

I hope what I write down is clear to you

Many thanks