Use the isomorphism theorem to determine the group

Albert Byrd

Albert Byrd

Answered question

2022-04-10

Use the isomorphism theorem to determine the group GL2RSL2(R). Here GL2(R) is the group of 2×2 matrices with determinant not equal to 0, and SL2(R) is the group of 2×2 matrices with determinant 1. In the first part of the problem, I proved that SL2(R) is a normal subgroup of GL2(R). Now it wants me to use the isomorphism theorem. I tried using
|GL2RSL2(R)|=|GL2(R)||SL2(R)|,
but since both groups have infinite order, I don't think I can use this here.

Answer & Explanation

cinereod3am

cinereod3am

Beginner2022-04-11Added 10 answers

Step 1
Define f: f:GL2(R)R such that
f(M)=det(M)Onto
Let M1,M2GL2(R) then
f(M1M2)=det(M1M2)=det(M1)det(M2)=f(M1)(M2)
Step 2
Hence f is homomorphism for ker(f)={M:MGL2(R)  such that  det(M)=1}=SL2(R)
From first theorem in isomorphism
GL2RSL2(R)R

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?