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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.

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.
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solution:- and ababal let a,bt G be such that order of b=2 Now, then, slabāk = baik clearly ababa acab)= alban a2b = abat = c

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