Question

A group has fewer than 100elements and subgroups of order of10and25. What is the order of $\mathit{G}$?

Short answer

The order of G is 50.

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 G

Let G be the group containing less than 100 elements and H and K be the subgroups of order 10 and 25.

The subgroup H of order 10 has the elements $\left\{10,20,30,40,50,60,70,80,90\right\}$ and the subgroup K of order 25 has the elements $\left\{25,50,75\right\}$.$\left\{25,50,75\right\}$.

The common element in the both the subgroup is 50.

Therefore, the order of G is 50.