- #1
Jason Bennett
- 49
- 3
- Homework Statement
- see title
- Relevant Equations
- see below
Progress:π:π(3)ββ€2π:π(3)βππ(3)π:π(3)/ππ(3)ββ€2
π(π)=det(π)
with πβπ(3), that way
π(π)β¦{β1,1}β β€2,
where 1 is the identity element.Ker(π) = {πβππ(3)|π(π)=1}=ππ(3), since det(π)=1 for πβππ(3).By the multiplicative property of the determinant function, π = homomorphism.
***What is the form of the canonical homomorphism (π) in this case?I'm used to the coset language,i.e. π:πΊβπΊ/Ker(π)
with π(π)=ππΎ for πΎ=ker(π)
If this were settled, then π is an isomorphism by the isomorphism theorem.
π(π)=det(π)
with πβπ(3), that way
π(π)β¦{β1,1}β β€2,
where 1 is the identity element.Ker(π) = {πβππ(3)|π(π)=1}=ππ(3), since det(π)=1 for πβππ(3).By the multiplicative property of the determinant function, π = homomorphism.
***What is the form of the canonical homomorphism (π) in this case?I'm used to the coset language,i.e. π:πΊβπΊ/Ker(π)
with π(π)=ππΎ for πΎ=ker(π)
If this were settled, then π is an isomorphism by the isomorphism theorem.