Question

Let $\mathit{H}$ and$\mathit{K}$ be subgroups of a finite group$\mathit{G}$ such that$\mathbf{[}\mathbf{G}\mathbf{:}\mathbf{H}\mathbf{]}\mathbf{=}\mathit{p}$ and$\mathbf{[}\mathbf{G}\mathbf{:}\mathbf{K}\mathbf{]}\mathbf{=}\mathit{q}$ , with$\mathit{p}$ and$\mathit{q}$ distinct primes. Prove that$\mathit{p}\mathit{q}$ divides $\mathbf{[}\mathbf{G}\mathbf{:}\mathbf{H}\mathbf{\cap }\mathbf{K}\mathbf{]}$.

Short answer

It is proved that, $pq$divides$[G:H\cap K]$ .

Step-by-step solution

Determine pq divides [G:H∩K]

Consider the given function, $H$and $K$be subgroups of a finite group $G$and $p$and$q$are distinct primes. Also, consider that $[G:H]=p$and $[G:K]=q$.

By applying Lagranges theorem,

$\left|G\right|=\left|H\right|[G:H]=\left|K\right|[G:K]=|H\cap K|[G:H\cap K]$

Put $[G:H]=p$and $[G:K]=q$in above equation.

$p\left|H\right|=q\left|K\right|=|H\cap K|[G:H\cap K]$

Further simplification of pq divides [G:H∩K]

Now, again applying to$H$and$H\cap K$respectively, then

$\left|H\right|=|H\cap K|[H:H\cap K]$ and $\left|K\right|=|H\cap K|[K:H\cap K]$

Hence from above two equations,

$p|H\cap K|[H:H\cap K]=q|H\cap K|[K:H\cap K]=|H\cap K|[G:H\cap K]$

Implies that,$p[H:H\cap K]=q[K:H\cap K]=[G:H\cap K]$

Hence both$p$ and$q$ divides$[G:H\cap K]$ ,as$p$ and$q$ are distinct primes. It follows that$pq$ divides$[G:H\cap K]$ .

Hence proved.