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Math Help - what is binary dihedral group?

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    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, \text{Aut}(C_{11}) \simeq C_{10} and so it is not dicyclic.
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  3. #3
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    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
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