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    Semi-Direct Products, S4 Groups and Homomorphisms

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    Let G = (Z/3Z)^4 SemiDirectProduct S_4 be the semi-direct product of (Z/3Z)^4 and S_4. Here S_4 acts on (Z/3Z)^4 by permutating the coordinates.

    Hint: Given H1, H2 an element in (Z/3Z)^4 and K1, K2 an element in S4. The semi-direct product is given by the operation (H1, K1) * (H2, K2) = (H1 + K1(H2), K1 * K2)

    A) Find the Center of G, Z(G).

    B) Let phi:(Z/3Z)^4 SemiDirectProduct --> {+- 1} be the map from G to Z/2Z given by phi (h,k) = sign(k). Show this map is a homomorphism.

    Check file for full problem. Please provide a clear solution step by step.

    © BrainMass Inc. brainmass.com April 1, 2020, 11:18 am ad1c9bdddf
    https://brainmass.com/math/group-theory/semi-direct-products-groups-homomorphisms-40771

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    Proof:

    A) We consider an element (h,k) in Z(G). Then for any (h1,k1) in G, we have
    (h,k)*(h1,k1)=(h1,k1)*(h,k)
    Then (h+k(h1),k*k1)=(h1+k1(h),k1*k)
    Then we have
    1) k*k1=k1*k, for any k1 in S4. This means that k is in Z(S4). But we know Z(S4)={e}.
    Thus we have k=e.
    2) h+k(h1)=h1+k1(h), since k=e, then we have h+h1=h1+k1(h), then h=k1(h) for any ...

    Solution Summary

    Semi-Direct Products, S4 Groups and Homomorphisms are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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