Answer to Question #166036 in Abstract Algebra for SHUNNY RAM YADAV

Question

Which of the following mapping are group homomorphisms? In case they are group homomorphisms find the kernel.

(i) f:R - {0} - R {0} under multiplication defined by f{0}=|x|,

(ii) f:z - R under addition defined by f(x) = x

Answer 1

*(i)*

 for x,y in R-{0},

f(x*y) = f(xy) = |xy| =|x||y| = f(x)*f(y).

And since f(x*y) = f(x)*f(y); the mapping f as defined f(x) =|x| is a group homomorphism under multiplication "*"

The kernel of the mapping f, ker(f) = { x in R-{0} such that f(x)=|x|= 0 in R{0} }.

So, ker(f) = {0}; since |x|=0 if and only if x=0

*(ii)*

 for x,y in Z,

f(x+y) = x+y = f(x)+f(y), the mapping f as defined f(x) = x is a group homomorphism under the addition "+".

The kernel of the mapping f, ker(f) = { x in Z such that f(x)=x= 0 in R }.

So, ker(f) = {0}; since x =0 if and only if x=0.