Problem about normal subgroups

[vikas92]

Homework Statement

If A,B and C are normal subgroups of G where B\subseteqA show that
A\bigcapBC=B(A\bigcapC)

Homework Equations

The Attempt at a Solution

Let x\inA\bigcapBC.then x\inA and x\inBC
Now as B\subseteqA thus BA=A.thus left side is BA\bigcapBC

Dont know how to proceed.

 

[micromass]

So you know that x\in A and x\in BC. The last means that there are b\in B and c\in C such that x=bc.

You must prove that x\in B(A\cap C). So you must write x as a product of something in B and something in A\cap C.

 

[vikas92]

x=bc \Rightarrow a^-1bac\inBC
Also a^-1bca\inBC Now a^-1b=a₁ for some a₁\inA\Rightarrowa₁ca\inBC
a₁(aa^-1)ca\inBC
(a₁a)c₁\inBC for some c₁\inC
a₂c₁\inBC
Cant still figure out