sub-groups

**x-laplace** · November 12th 2009, 12:02 PM

Is $G$ a finite group , $H$ $,$ $K$ subgroups such that $H K$ .
Prove $[ G:H ] = [ G:K ] . [ K:H ]$

Please can develop it in detail.

THANKS YOU FRIENDS....

**Jose27** · November 12th 2009, 02:09 PM

Its just a simple application of Lagrange:

$[G:H]=\frac{ \vert G \vert }{ \vert H \vert }$ and $\vert H \vert = \frac{ \vert K \vert }{ [K:H] }$ substitute in the first and you get $[G:H] = \frac{ \vert G \vert }{ \vert K \vert } [K:H] = [G:K][K:H]$

**x-laplace** · November 12th 2009, 09:34 PM

thanks for the help, that I needed to think a bit

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