- $(Nx)(Ny)=Nxy$
- $N(x)(Nx^{-1})=(Nx^{-1})(Nx)=N$
- $N(Nx)=(Nx)N=Nx$
Deduce that the right cosets of $N$ in $G$ form a group under the given multiplication.
Deduce that the right cosets of $N$ in $G$ form a group under the given multiplication.
Something which I like to do is using the monoid of nonempty subsets of $G$. Let $A,B\subset G$, we will define $AB=\{ab|a\in A,b\in B\}$. It is easy to show that multiplication of sets is associative. Note that the cosets $Nx,xN$ are $N\{x\},\{x\}N$ respectively. Also note that for any subgroup of $H$ of $G$ we have $HH=H$ In the next proof I will use the facts :$NN=N$ ,$N\{x\}=\{x\}N$ and the fact that multiplication of subsets is associative.
a) $(N\{x\})(N\{y\})=N(\{x\}N\{y\})=N(N\{x\}\{y\})=NN(\{x\}\{y\})=N\{xy\}$
b),c) can be proved similarly.