- #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##,
$$(G/H)/(K/H) \simeq (G/K)$$