Group Homomorphisms for General Linear Groups I am doing some revision for upcoming exams and as a practice I am trying to find an example of a Group Homomorphism for the following domains and codomains:
A) $GL_4(\mathbb{R})  \to  GL_1(\mathbb{R})$
B) $GL_3(\mathbb{R})  \to  GL_4(\mathbb{R})$
C) $GL_3(\mathbb{R})  \to  SL_3(\mathbb{R})$
D) $GL_3(\mathbb{R})  \to  GL_2(\mathbb{R})$
I know the answer to A) is the determinant, and I'm pretty sure that part d) isn't possible, but could someone please help me with the other two?
Thanks!
 A: I'm going to cheat a bit and generalise the answer for part A.
For parts A, B, and D, each matrix $M$ maps to a diagonal matrix where all diagonal entries are the determinant of $M$.
For part C, offhand I think you can take each matrix $M$ mapping to a scaled version of $M$, that is, $kM$ where $k\in\mathbb{R}$, so that $\det (kM) = 1$.
A: For B), get intuition by moving from $2$-space to $3$-space; if we have a two-dimensional rotation matrix $\pmatrix{a & b \\ -b & a}$, we can do the same operation but in $3$ dimenions by fixing the $z$-axis. The map
$$M = \pmatrix{a & b \\ -b & a} \mapsto \pmatrix{a & b & 0 \\ -b & a &0 \\ 0 &0 &1} = \pmatrix{M & 0 \\ 0 & 1} \in GL_3(\Bbb R)$$ works, and will work for general transformations (not just rotations) to hop up as many dimensions as you like.
You can do something similar in D), by transforming in $3$-space then projecting onto the $xy$-plane and ignoring what happens to the $z$-coordinate to go from $GL_3(\Bbb R)$ to $GL_2(\Bbb R)$.
The canonical map in C) is the map that sends a matrix $M$ to $\frac{1}{\det M}M$.
