How to show that these groups are isomorphic? Show that group of all real matrices of form [[x,y],[-y,x]], (x,y) ne (0,0)

babuliaam

babuliaam

Answered question

2022-09-26

How to show that these groups are isomorphic?
Show that group of all real matrices of form
[ x y y x ] , ( x , y ) ( 0 , 0 )
is isomorphic with/to C∖{0} under complex multiplication?
I know two ways to show isomorphism: 1) finding a homomorphic function 2) writing the multiplication table and comparing.

Answer & Explanation

Absexabbelpjl

Absexabbelpjl

Beginner2022-09-27Added 8 answers

Step 1
What's behind this exercise is the following.
Consider C as a vector space over R, with basis 1,i. For each x + i y C , with x , y R , consider the map
C C z z ( x + i y ) .
Step 2
This map is R-linear, and its matrix with respect to the basis 1,i is precisely
[ x y y x ]
See if you can get on from here.
Note. The matrix is the one above if you consider row vectors. If you consider column vectors, then take the map z z ( x i y ), which is still R-linear.
sombereki51

sombereki51

Beginner2022-09-28Added 3 answers

Step 1
Let G be the group of those matrices with usual matrix multiplication. Define
f : G C { 0 }
Step 2
Show that it is an isomorphism.

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