Question 32850: PLEASE NEED YOUR HELP
Which of the following functions are homomorphisms?
a) f: Z (right arrow), Z defined by f(x)= -x
b) f: Z2 (right arrow), defined by f(x)= -x
c) g: Q (right arrow), defined by g(x) = (1/(x^2+1))
d) h: R (right arrow), M2x2(R), defined by h(a)= {(-a,a) , (0,0)}
e) f: Z12 (right arrow) Z4, defined by f([x]12) = [x]4
Thank You
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! You really did not know how to type --> , did you ???
a ) f: Z --> Z defined by f(x)= -x
b) f: Z2 -->Z2, defined by f(x)= -x
Clearly a,b) are additive homo. (too simple)
c) g: Q --> Q defined by g(x) = (1/(x^2+1))
since g(1) = 1/2 ~= 1 , so g cannot be a multiplicative homo
[ Note g(1) must be equal to 1 ]
d) h: R --> M2x2(R), defined by h(a)= {(-a,a) , (0,0)}
check h(a+b) = ( -(a+b) , (a+b))
(0 , 0)
= (-a , a) + (-b, b)
(0 , 0) (0, 0)
= h(a) + h(b)
So, h is a homo from (R, +) to (M2x2(R), +)
e) f: Z12 --> Z4, defined by f([x]12) = [x]4
First of all, you have to know that f is weel-defined, i.e.
[x]1= [y]12 --> x = y mod 12 --> x = y mod 4 --> [x]4 = [y]4
f([x]12+ [y]12) = f([x+y}12) = [x+y]4 = [x}4+ [y]4 = f([x]12)+ f([y]12)
So, f is a homo. from Z12 to Z4.
Of course, you have to work hard.
Kenny
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