Question

What are the possible orders of the subgroup of $\mathit{G}$when $\mathit{G}$ is

b)${\mathbf{\mathbb{Z}}}_{\mathbf{24}}$

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 $\left\{0,1,2,\dots ,23\right\}$ 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 $\left\{1,2,3,4,6,8,12,24\right\}$ which is of order 8.

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