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Math Help - mapping homomorphism and subgroup

  1. #1
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    mapping homomorphism and subgroup

    1) Describe explicitly all homomorphisms h: C_6 maps to Aut(C_12)

    Firstly, I have to find what group Aut(C_12) is isomorphic to, and then find the order of that group. Is this the right track?

    If it is, then I don't know how to do after this.

    How do you find the homomorphism?

    2) Find a subgroup H of sigma_8 ( permutation of {1,2,.......,8} ) such that H =~ Q8 ( H is isomorphic to the quanterion group of 8)

    What should I do for this question?


    Thanks alot to all your help.
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    1) Describe explicitly all homomorphisms h: C_6 maps to Aut(C_12)

    Firstly, I have to find what group Aut(C_12) is isomorphic to, and then find the order of that group. Is this the right track?

    If it is, then I don't know how to do after this.

    How do you find the homomorphism?
    Hint: \text{Aut}(C_{12}) \simeq \mathbb{Z}_{12}^{\times} \simeq \mathbb{Z}_2\times \mathbb{Z}_2.

    2) Find a subgroup H of sigma_8 ( permutation of {1,2,.......,8} ) such that H =~ Q8 ( H is isomorphic to the quanterion group of 8)
    Find elements a,b with |a|=|b|=4 and a^2 = b^2 and then bab = a^{-1}. Once you do that the subgroup generated by these elements will satifisy the conditions of being isomorphic to the quaternion group.
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