I assume you meant ##kh_4h_3 \in K/H## and not ##kh_4h_3 \in KH## at the end there.
Now,
##\Leftarrow##
In an exam I would put a lot more effort into initializeg the different variables ##k_i##, ##g_i## and ##h_i## and justifying each step.
If ##K/H \triangleleft G/H##, we have that...
I am still unsure how to go about resolving this problem. I haven't really seen any examples of problems with quotients like this one. So it's the possible manipulations that I am unsure about. Please tell me if this is alright.
This problem is an iif so I need to demonstrate both directions...
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...
Homework Statement
So I have these two Matrices:
M = \begin{pmatrix}
a & -a-b \\
0 & a \\
\end{pmatrix}
and
N =
\begin{pmatrix}
c & 0 \\
d & -c \\
\end{pmatrix}
Where a,b,c,d ∈ ℝ
Find a base for M, N, M +N and M ∩ N.
Homework Equations
I know the 8 axioms about the vector spaces.
The...