1. ## maximal normal subgroup

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

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

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

1) $\displaystyle H < K$

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

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

Tonio

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

1) $\displaystyle H < K$

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

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