If a g and a12 e what arc the possibilities for the order o

If a g and a^12 = e, what arc the possibilities for the order of a? (b) If b C is a nonidentity element and b^p = e for a prime p, show that b must have order p.

Solution

We know that the order of an element a of a group is the smallest positive integer n such that an = e (where e denotes the identity element of the group). Thus, if a G .and a12 = e, then the order of a must be a factor of 12, i.e. the order, n, of a must be 1,2,3,4 6 or12. If b G is a non-identity element and bp = e for a prime p, then, by the definition of the order of an element of a group, the order of b must be a factor of p. However, since p is a prime, therefore its only factors are 1 and p. Further b1 = b e as b is a non-identity element of G. Hence the order of b must be p.