# Recent content by Alex Langevub

1. ### An exercise with the third isomorphism theorem in group theory

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...
2. ### An exercise with the third isomorphism theorem in group theory

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...
3. ### An exercise with the third isomorphism theorem in group theory

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...
4. ### Is the zero Matrix a vector space?

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...