Question

Let $\square **$be the multiplicative group of positive real numbers. Show that $f:\square **\to \square **$given by $f\left(x\right)=3x$is not a homomorphism of groups

Short answer

The function$f:\square **\to \square **$ defined by$f\left(x\right)=3x$ is not a homomorphism

Step-by-step solution

Definition of Isomorphism

Let G and H be the groups with group operation denoted by **. Group G is isomorphic to group H if there is a function $f:G\to H$such that:

(i) f is injective

(ii) f is surjective

(iii) $f\left(ab\right)=f\left(a\right)f\left(b\right)$for all$a,b\in G$

f is injective

Let $a,b\in \square **$such that $f\left(a\right)=f\left(b\right)$implies,

$$\begin{gathered} f\left(a\right)=f\left(b\right) \\ 3a=3b \\ a=b \end{gathered}$$

Hence, f is injective.

f is surjective

Let $a\in \square **$in co-domain then, $f\left(a\right)=b$such that$3a=b$implies $a=\frac{b}{3}$.

Thus, for any real number a in $\square **$there exists an element $\frac{b}{3}in\square **$such that

$$\begin{gathered} f\left(a\right)=3a \\ =3\left(\frac{b}{3}\right) \\ =b \end{gathered}$$

Hence, f is surjective.

f is a homomorphism

Let$a,b\in \square **$ then,

$$\begin{gathered} f\left(ab\right)=3\left(ab\right) \\ \ne f\left(a\right)f\left(b\right) \end{gathered}$$

Thus, for all $a,b\in \square **,f\left(ab\right)\ne f\left(a\right)f\left(b\right)$, and hence, f is not a homomorphism.

Therefore, the function $f:\square **\to \square **$defined by$f\left(x\right)=3x$ is not a homomorphism