- #1

Alex Langevub

- 4

- 0

## Homework Statement

Let ##G## be a group. Let ##H \triangleleft G## and ##K \leq G## such that ##H\subseteq K##.

a) Show that ##K\triangleleft G## iff ##K/H \triangleleft G/H##

b) Suppose that ##K/H \triangleleft G/H##. Show that ##(G/H)/(K/H) \simeq G/K##

## Homework Equations

The three isomorphism theorems.

## The Attempt at a Solution

a) Let ##k \in K##, ##h \in H## and ##g \in G##

##(gH)(kH)(gH)^{-1}## need to belong to ##kH##

I am not quite sure how to go about proving this.

b) Since ##K/H \triangleleft G/H##, we know that ##K\triangleleft G## as proven in a).

We can therefore apply the third isomorphism theorem which states that since ##H## and ##K## are both normal in ##G## and that ##H## is a subset of ##K##,

$$(G/H)/(K/H) \simeq (G/K)$$