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

ID: 2986023 • Letter: L

Question

Let G = <[I, A, B, C, D, K}, Matrix Multiplication> R*= <x is a real number, where x cannot equal 0}, x> 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!

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