You probably mean . If is the order of then . So cannot be even, i.e. it must be odd.

3.The order of ab is the same as the order of ba. (Hint: if (ba)n = (baba…..b)a = e. Let (baba….b) =x then a is the inverse of x. Thus, ax =e.)

But thus .

(Why is this not a complete proof?)

Use induction, with the basic case covered in #2.2. Let x =a1a2… an, and let y be a product of the same factors, permuted cyclically. (That is, y = akak+1…ana1…ak-1.) Then ord(x) = ord(y).

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