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Semi-Direct Products, S4 Groups and Homomorphisms

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.


Solution Preview


A) We consider an element (h,k) in Z(G). Then for any (h1,k1) in G, we have
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 ...

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