Give an explicit isomorphism . Solution Let A = Z11 , x B = Z10 , + f:B -> A f(b

Question

Explanation / Answer

Let A = Z11 , x B = Z10 , + f:B -> A f(b) = 2^b (mod 11) its an isomorphism f(b + c) = f(b)f(c) which is true as 2^(b+c) = 2^b * 2^c also kernel of f = {b| f(b) = 1(identity in A)} = {0} As 2^x = 1 (mod 11) means that x=0 (mod 10) as 2 is a primitive root mod 11 And f is onto as 2 is a primitive root mod 11 (if you do not know about primitive roots just list out the entire function and verify that f is indeed one one and onto) f(0) = 1; f(1) = 2; f(2)=4; f(3) = 8; f(4) = 5; f(5) = 10; f(6) = 9; f(7)=7; f(8)=3; f(9)=6; Hence f is one one and onto and a isomorphism Hence proved please Rate.

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.