# Let a and b be elements of a group G such that b has order 2...

Let a and b be elements of a group G such that b has order 2 and ab=ba^-1 12. Let a and b be elements of a group G such that b has order 2 and ab = ba-1. (a) Show that a” b = ba-n for all integers n. Hint: Evaluate the product (bab)(bab) in two different ways to show that ba+b = a-2, and then extend this method. (b) Show that the set S = {a”, ba" | n € Z} is closed under multiplication and in fact forms a group. (c) Show that S (a,b), the dihedral group with generators a and b.  ##### Add Answer of: Let a and b be elements of a group G such that b has order 2...
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• ### 1. Suppose a and b are elements of a group G. Prove, by induction, (bab−1)n = banb−1 . Hence prove that if a has order m, then bab−1 also has order m. Deduce from question (#1) that in any group ab an...

1. Suppose a and b are elements of a group G. Prove, by induction, (bab−1)n = banb−1 . Hence prove that if a has order m, then bab−1 also has order m. Deduce from question (#1) that in any group ab and ba have the same order (you may assume ab has finite order). The assertion in Question (#1) can be generalized to an assertion about isomorphisms. State and prove it.

• ### 1. Let a and b be elements of a group . Prove that ab and ba... 1. Let a and b be elements of a group . Prove that ab and ba have the same order. 2. Show by example that the product of elements of nite order in a group need not have nite order. What if the group is abelian?

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• ### Group Theory 2

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• ### Let D4 be dihedral group order 8. So D4={e, a, a^2, a^3, b, ab, a^2b, a^3b},   a^4 = e,...

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• ### dihedral group Dn

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• ### Let G be a group, and let a ∈ G. Let φa : G −→ G... Let G be a group, and let a ∈ G. Let φa : G −→ G be defined by φa(g) = aga−1 for all g ∈ G. (a) Prove that φa is an automorphism of G. (b) Let b ∈ G. What is the image of the element ba under the automorphism φa? (c) Why does this imply that |ab| = |ba| for all elements a, b ∈ G? 9. (5 points each) Let G be a group, and let...

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