Question

The Number of Elements Satisfying g7=e in a finite Group F is ______

(a) even

(b) not a number

(c) odd

(d) rational

I got this question in an internship interview.

The above asked question is from Cyclic Groups in chapter Groups of Discrete Mathematics

Answer 1

The correct answer is (c) odd

For explanation I would say: Let g≠e be an element in group F such that g7=e. As 7 is a prime number, this yields that the order of g is 7. Consider, the subgroup (g) is generated by g. As the order of g is 7, the order of the subgroup (g)  is 7. Hence, the order must be odd.