Originally Posted by
kathrynmath Show that C^x is isomorphic to the subgroup of GL2(R) consisting of all matrices of the form:
a b
-b a such that a^2+b^2 is not 0.
I'm guessing I need to show C^x--->G given by c(a+bi)=
a b
-b a is an isomorphism.
I need to show 1-1, onto, and homomorphism.
1-1:
Assume c(a+bi)=c(c+di). I need to show this implies a+bi=c+di
a b c d
-b a = -d a
I'm not sure how this implies 1-1
onto:Not sure how to show this
homomorphism:
c(ab)=c(a)c(b)
Let a=a+bi, b=c+di
a b c d
-b a * -d c
ac-db ad+bc
-bc-ad -bd+ac =c(ab)