Problem about normal subgroups

[vikas92]

Homework Statement

If A,B and C are normal subgroups of G where B$\subseteq$A show that
A$\bigcap$BC=B(A$\bigcap$C)

Homework Equations

The Attempt at a Solution

Let x$\in$A$\bigcap$BC.then x$\in$A and x$\in$BC
Now as B$\subseteq$A thus BA=A.thus left side is BA$\bigcap$BC

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$\in$BC
Also a^-1bca$\in$BC Now a^-1b=a₁ for some a₁$\in$A$\Rightarrow$a₁ca$\in$BC
a₁(aa^-1)ca$\in$BC
(a₁a)c₁$\in$BC for some c₁$\in$C
a₂c₁$\in$BC
Cant still figure out