Let G = <[I,A,B,C,D,K} Matrix Multiplication> R*= <x is a real number, where x c

ID: 2986090 • Letter: L

Question

Let G = <[I,A,B,C,D,K} Matrix Multiplication> R*= <x is a real number, where x cannot equal zero>

R* is all positive real numbers

Define f: G --> R* by f(X) (the determinant of X) for any matrix X in G

1. Find all conjugates of A: ______ all conjugates of B: ________

2. f is a Homomorphism: G---> R*. Why?

3. List all elements in the Kernel of f? Ker(f)={________}

4. List all elements in the homomorphic image of G, i.e. the Range(f).

Im(G)= Ran(f)= {_________}.

5. Let K= Ker(f). (G:K)=__________

6. List the elements of each coset of K in G.

K__={_____} K__={_____} K__={_____}

7. Complete both mulitplication tables

coset multiplication table for quotient group G/K

mulitplication table for Im(G)

Please be detailed and I will reward full points! :) Thanks!

(1)conjugates of A ,,,A,BAB-1 ,CAC-1 ,DAD-1,KAK-1

conjugates of B,,,B,ABA-1,CBC-1,DBD-1,KBK-1.........

-1 means its inverse................

(2)we know that det(AB)=det(A)*det(B).......f is determinant.....hence f is homomorphism

(3)ker(f) is the set containing matrices whose determinant is 1...........hence I is an element of ker(f)

(4)Ran(f)={1,det(A),det(B),det(C),det(D),det(K)}

(5)KA,K,KB,KC,KD,K^2

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