1. ## maximal normal subgroup

Prove that a a normal subgroup $H$ of the group $G$ is maximal if and only if the Quotient Group $\frac{G}{H}$ is simple.

2. Originally Posted by sashikanth
Prove that a a normal subgroup $H$ of the group $G$ is maximal if and only if the Quotient Group $\frac{G}{H}$ is simple.

Remember the correspondence theorem for groups: every subgroup of the quotient group $G/H$ is of the form $K/H$ , where $K$ is a subgroup of $G$ , with the following characteristics:

1) $H < K$

2) $[G/H:K/H]=[G:K]$

3) $K/H$ is normal in $G/H$ iff $K$ is normal in $G$

Tonio

3. Originally Posted by tonio
Remember the correspondence theorem for groups: every subgroup of the quotient group $G/H$ is of the form $K/H$ , where $K$ is a subgroup of $G$ , with the following characteristics:

1) $H < K$

2) $[G/H:K/H]=[G:K]$

3) $K/H$ is normal in $G/H$ iff $K$ is normal in $G$