Exercise 3.1. Let G be a group. Given g € G, define ig : G → G by ig(x)=gxg, Vx Є G. (1) Show that i, is an isomorphism of G with itself (i.e. an automorphism of G). (2) Show that CG(g) = {x G | ig(x) = x} = {x = G | xg = gx} is a subgroup of G, called the centralizer of g in G.

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

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Q: 5.3 Let H be the set of elements () of GL(2, R) such that ad-bc-1. Show that H is a subgroup of…

A: be the set of elements  of  such that  that is the We have to show that  is a subgroup of

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Q: Let G be a group. Define Z(G) = {x G | xg = gr for all g € G}. Prove Z(G) ≤ G. Find Z(S3) and Z(D4).

A: Solution:Let  G be a group. Define Clearly, eSuppose be  arbitrary.Then,  [since, x,y]Thus, 2nd…

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Q: 5. True or False. Explain your answers. (a) Suppose f: G → H is a homomorphism and g € G. Then, the…

A: We have to answer that statements are true or false.

Q: Let G be a group and H ≤ G. (a) Prove for any a € G, aHa-¹ ≤G. (b) Let x E G and Prove C (x) ≤ G.…

A: Vital instruction : According to Bartleby Guideline , i can solve at most first three subparts if…