Question

Assume that G is a group, H is a subgroup of G, |H|=6, [G:H]>4 and |G|<50. What can be concluded about |G|?

Answer 1

Let $G$ be a group, $H$ be a subgroup of $G$, $|H|=6, [G:H]>4$ and $|G|<50$.  According to Lagrange's theorem, $|G|=|H|\cdot[G:H]=6\cdot [G:H]>6\cdot 4=24$. Therefore, we conclude that

$24<|G|<50.$