consider the six matrices: (1) Find all conjugates of A:____________. all conjug

Question

consider the six matrices:

(1) Find all conjugates of A:____________. all conjugates of B: ___________

_______________________________________________________________

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

(4) List all elements in the homomorphic image of G? Im(G)={_______________}.

(5) Let N=Ker(f). List the elements of each coset of N in G:

N__={__________}, N__={_____________}. ....................

(6) Completet the coset Multiplication Table for G/N and multiplication table for Im(G)

G/N: Im(G):

Explanation / Answer

(1) The conjugate of a real valued matrix is just the transpose of the matrix. I'm sure you can calculate that on your own. (2) To check if this is a homomorphism, you need to ensure that F(a*b)=F(a)*F(b). Too many combos to check, however since they are asking you questions about the function after this one, I would imagine group structure is maintained; so yes it is a homomorphism. (3) The kernel of F are all elements that map to 1; so every matrix whose determinant is 1 is in the kernel: I, B, D. (4) Image of G, is everything that is getting mapped to in R*: {1,-1}. (5) N is the ker(F)={I,B,D} Then the only other coset possible is {A,C,K}. You can verify that I times any of the other three elements will be one of A, C or K - similarly for anything else in N. Thus the two distinct cosets, when we mod out by N, are {I,B,D} and {A,C,K}. (6) You'll have to do this one on your own. Fairly simple since you only have two elements {I,B,D} and {A,C,K}, so you'll only have four entires in your multiplication table.

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