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Abstract Algebra Show that On is generated by { (1, 2), (1, 2, 3, ...n)) . [Hint Show that as r varies, (1, 2, 3, ...)" (1, 2) (1, 2,3, ..., n)" gives all the transpositions (1, 2), (2, 3), (3, 4)... (n - 1, n), (n, 1). Then show that any transposition is a product of these transpositions and use the following theorem]. Theorem: Any permutation of a finite set containing at least two elements is a product of transpositions. Step 1 : Compute the first few transpositions So when, r=0 would get (1,2) when r=1 would get (2,3) when r=2 would get (3,4) if keep going if r=n-1 would get (n,n+1) I am not sure how to obtain the product to obtain (1,n) In essence, going with the pattern above would have a general format being (1, 2, 3, ..., n)" (1, 2) (1, 2, 3, ..., n)" = (r+1,r+ 2 I know, that n is fixed, so that no matter what n is would get the same transposition as r varies. Just not sure how to obtain (1, ") as not sure what r would be? Step 2: Prove the problem by the use of matematical induction