Question

A finite group G of order 219 is __________

(a) a semigroup

(b) a subgroup

(c) a commutative inverse

(d) a cyclic group

This question was addressed to me in an internship interview.

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

Answer 1

The correct option is (d) a cyclic group

Easy explanation: The prime factorization 219=3⋅73. By the definition of Sylow’s theorem, determine the number np of Sylow p-group for p=3,73. np≡1(mod p) and np divides n/p. Thus, n3 could be 1, 4, 7, 10, 13,… and n3 needs to divide 219/3=73. Hence the only possible value for n3 is n3=1. So there is a unique Sylow 3-subgroup P3 of G. By Sylow’s theorem, the unique Sylow 3-subgroup must be a normal subgroup of G. Similarly, n73=1, 74,… and n73 must divide 219/73=3 and hence we must have n73=1. Thus, G has a unique normal Sylow 73-subgroup P73.