Can You Prove These Properties of Normal Subgroups in Group Theory?

[cbarker1]

Dear Everyone,

I am struck on a problem dealing with normal subgroups.

The problem is the following:

Let $G$ be a group, $H$ and $K$ be normal subgroups of $G$ with $H\ge K$.

1. Prove that $H$ is a normal subgroup of $K$
2. Prove that $K/H$ is a normal subgroup of $G/H$.

My attempted work:

Proof: We know that $H,K\ge G$.

Thanks
Carter B

 

[Opalg]

The problem is the following:

Let $G$ be a group, $H$ and $K$ be normal subgroups of $G$ with $H\ge K$.

1. Prove that $H$ is a normal subgroup of $K$
2. Prove that $K/H$ is a normal subgroup of $G/H$.

It looks as though $H\ge K$ should be $H\le K$. The rest of the question certainly implies that $H$ is contained in $K$.

Start by writing down the definition of a normal subgroup. Use that definition to say what it means for $H$ to be a normal subgroup of $G$. You should be able to deduce the result 1. from that.

For 2., again start by writing down some definitions. How is a quotient group defined? Then what conditions does $K/H$ have to satisfy in order to be a normal subgroup of $G/H$?

 

[Olinguito]

Dear Everyone,

I am struck on a problem dealing with normal subgroups.

The problem is the following:

Let $G$ be a group, $H$ and $K$ be normal subgroups of $G$ with $H{\color{red}\le} K$.

1. Prove that $H$ is a normal subgroup of $K$
2. Prove that $K/H$ is a normal subgroup of $G/H$.

My attempted work:

Proof: We know that $H,K{\color{red}\le} G$.

Thanks
Carter B

Hi Carter.

Note the corrections in red.

1. To show that $H$ is a normal subgroup of $K$, you need to snow that $kHk^{-1}=H$ for all $k\in K$. But $H$ is a normal subgroup of $G$, i.e. $gHg^{-1}=H$ for all $g\in G$. Can you complete the proof?2. $K/H$ is the set of all cosets of $H$ in $K$, i.e. cosets of the form $kH$ for $k\in K$. An element of $G/H$ is a coset of the form $gH$ where $g\in G$. To show that $K/H$ is a normal subgroup of $G/H$, you need to show that
$$\left(gH\right)\left(kH\right)\left(gH\right)^{-1}$$
is in $K/H$. The above is equal to
$$\left(gkg^{-1}\right)H.$$
As $K$ is normal in $G$, what can you say about $gkg^{-1}$?