Question

The order of a simple abelian group is __________

(a) infinite

(b) real number

(c) finite

(d) prime

This question was posed to me in an interview for internship.

My question is based upon Cyclic Groups topic in division Groups of Discrete Mathematics

Answer 1

The correct option is (a) infinite

Explanation: Let p be the order of g (hence the order of G). As a contradiction, assume that p=ab is a composite number with integers a > 1, b > 1. Then (ga) is a proper normal subgroup of G. This is a contradiction since G is simple. Thus, p must be a prime number.

Therefore, the order of G is a prime number.