Question

What are the possible orders of the subgroup of G when G is

(b)${\mathrm{\mathbb{Z}}}_{24}$

Short answer

The possible order of the subgroup of G is 8.

Step-by-step solution

Lagrange’s theorem

If K is a subgroup of a finite group G , then the order of K divides the order of G .

Order of subgroup of  G

The elements in the group ${\mathrm{\mathbb{Z}}}_{24}$are {0,1,2,.....,23} implies that the order of G is 24.

Let K be the subgroup of the finite group ${\mathrm{\mathbb{Z}}}_{24}$ then, the order of K divides the order of G .

By the definition, the order of K divides the order of G . Hence, all the divisors of order of G is the order of K .

The divisors of 24 are {1,2,3,4,6,8,12,24} which is of order 8.

Therefore, the possible order of subgroup of G is 8.