Hi, I need help solving these problems:
Let a, b, and c be elements of a group G. Prove the following:
1. If ak =e where k is odd, then the order of a is 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.)
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).