# Homework Help: Problem about normal subgroups

1. Jun 6, 2012

### vikas92

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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.

2. Jun 6, 2012

### 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 $A\cap C$.

3. Jun 6, 2012

### 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