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Groups and Representations

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Please help answer the following algebra questions. Provide step-by-step solutions.

(i) Now let G be a group with the presentation
G=<a,b|a^7=e,b^3=e,b^(-1) ab=a^2>
You are told that |G|=21. Let w=exp(2i*pi/(7)) E C. Prove that there is a representation
p:G --> GL(3,C)
with

p(a)= (w^2, 0, 0; 0, w^4, 0; 0, 0, w) and p(b) = (0, 1, 0; 0, 0, 1; 1, 0, 0)

(ii) For the representation p:G --> GL(3,C) defined in (i) above, let p*:G-->GL(3,C) be the representation given by p*(g)=p(g^(-1))^T for all gEG. Write down p*(a) and p*(b). Are p and p* equivalent representations?

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This solution helps with questions involving groups and representations. Step-by-step calculations are given, along with detailed explanations.

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(i) Now let G be a group with the presentation
G=<a,b|a^7=e,b^3=e,b^(-1) ab=a^2>
You are told that |G|=21. Let w=exp(2i*pi/(7)) E C. Prove that there is a representation
p:G --> GL(3,C)
with

p(a)= (w^2, 0, 0; 0, w^4, 0; 0, 0, w) and p(b) = (0, 1, 0; 0, 0, 1; 1, 0, 0)

Solution:
If ...

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