Answer to Question #162935 in Abstract Algebra for K

Question

Find a binary function * on Q such that phi is an isomorphism mapping <Q, +> with <Q , *) and give the identity element for *. The map is phi : Q --> Q is given by phi(x)=3x-1 for x is an element of Q

Answer 1

Solution:

Given an isomorphism mapping, i.e., one-to-one and onto map $\phi: {Q} \rightarrow {Q}$ defined as $\phi(x)=3 x-1$ .

Note that for any $x \in {Q}$ , we have $\phi\left(\frac{x+1}{3}\right)=x$

Here we find out what $a * b$ equals.

From the above line, and from the fact that the condition $\phi(x) * \phi(y)=\phi(x+y)$ must hold, we have,

$a * b=\phi\left(\frac{a+1}{3}\right) * \phi\left(\frac{b+1}{3}\right)=\phi\left(\frac{a+1}{3}+\frac{b+1}{3}\right)=\phi\left(\frac{a+b+2}{3}\right)=3( \frac{a+b+2}{3})-1=a+b+1$

Therefore our binary operation is $a * b=a+b+1$ .

For this particular binary operation the element $-1 \in {Q}$ is the identity because $-1 * a=-1+a+1=a$ .