(Solved): D. Further Elementary Consequences of Lagranges Theorem Let G be a finite group, and let H and K ...

D. Further Elementary Consequences of Lagrange’s Theorem Let G be a finite group, and let H and K be subgroups of G. Prove the following: 1 Suppose H ⊆ K (therefore H is a subgroup of K). Then (G: H) = (G: K)(K: H). 2 The order of H ∩ K is a common divisor of the order of H and the order of K.