# 代數學-請高手回答

@:Q-->Q defined by @(x)=5x-1 for x€Q is one to one and onto Q.Give the definition of a binary operation * on Q such that @ is an isomorphism mapping <Q, .(一點)>onto<Q,*>and then give the identity element for * on Q.

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• Anonymous

(1) a * b =( 1/5)(a+1)(b+1)

(2) The identity for * on Q is 4.

Solutions:

(1)At the first, try to find the definition of the binary operation* on Q for

which @ is an isomorphism.

Since @ is an isomorphism, so @ (x.y) =@(x) * @(y) ,

Hence, 5xy-1=(5x-1) *( 5y-1)----(#).

Let a=5x-1, b=5y-1, then x=(1/5)(a+1), y=(1/5)(b+1).

Replace x and y in terms of a and b in the equation(#),

we got 5(1/5)(a+1)(1/5)(b+1)=a*b, therefore we should define

a*b= (1/5)(a+1)(b+1)

so that @ is an isomorphism.

(2)Next, try to find the identity element for * on Q.

If b is the identity , then a*b=a, for all a in <Q,*>.

This leads to (1/5)(a+1)(b+1)=a, from this equation of b,

we got b = 4.

Therefore, the desired identity for * on Q is 4.

(3)

In order to complete the proof, you should check yourself that

we defined

a * b =( 1/5)(a+1)(b+1)

in this way is indeed a binary operation on Q .

Also, check that 4 is the desired identity for * on Q .

Anyway, these are easy routine works.