Definition 14.4.12. The order of a permutation 7 is the smallest positive integer k such that * = id. As before, the order of T is denoted by the notation |T|. A Proposition: Let 7 be a permutation,...

Please do exercise 14.4.13 please show step by step and explain.

The hint states "The first blank should be replaced by k"

Extracted text: Definition 14.4.12. The order of a permutation 7 is the smallest positive integer k such that * = id. As before, the order of T is denoted by the notation |T|. A Proposition: Let 7 be a permutation, and let k = |7|. Then r = id if and only if mod(e, k) = 0. Exercise 14.4.13. Fill in the blanks with the appropriate variables in the following proof of the proposition. (*Hint*) Proof: For any integer l we may write l = ak + b, where b e Z <1> . It follows that =7>k+<> = (7<4>-k)p<5> = (7k)<>>Z> = (id)<>,> =7 10>. Therefore 7 < 12=""> < k,="" and="" we="" also="" know="" that="">< 13=""> is the smallest positive integer such that 714> = id. Hence it must be the case that b = < 15="">, which is the same thing as saying that mod(e, < 17=""> ) = 0. = id if and only if r11> = id. However, we know that