Answer to Question #203312 in Abstract Algebra for Raghav

Question

If G is a group with o(g) < 100 and G has subgroups of order 10 and 25, what is the

order of G?

Answer 1

According to Lagrange's theorem for a finite group $G$, the order of every subgroup $H$ of $G$ divides the order of $G$. Let $n$ be the order of  $G$. Then $10$ divides $n$ and $25$ divides $n$. It follows that $50=LCM(10,25)$ divides $n.$ Taking into account that $n<100,$ we conclude that $n=50.$

Answer: the order of $G$ is 50.