# Math Help - Def of normal subgroup

1. ## Def of normal subgroup

The definition of a normal subgroup is: for each element, n, in N and each g in G, the element gng−1 is still in N where N is a subgroup of G. Can this be changed to ...element g−1ng is still in N. I think it still makes sense since gng−1=m for some m in N thus g−1gng−1g=g−1mg which gives n=g−1mg . Now g−1mg is in N since n is in N.
Is this correct?

2. You sir are correct.

Normally you write a subgroup N is normal in G iff $gNg^{-1}\subset N$ for all $g\in G$. This is your definition given, but you can also notice that $g\in G \Rightarrow g^{-1}\in G$ so this definition also tells you:
$(g^{-1})N(g^{-1})^{-1}=g^{-1}Ng\subset N$

Which is the second conclusion you drew.