Question

Prove that the function$g:{\square }_9\to {\square }_9$ defined by$g\left(x\right)=2x$ is an isomorphism.

Short answer

The function$g:{\square }_9\to {\square }_9$ defined by$g\left(x\right)=2x$ is an isomorphism.

Step-by-step solution

Definition of Isomorphism

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

(i) f is injective

(ii) f is surjective

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

g is injective

Let $a,b\in {\square }_9$such that $g\left(a\right)=g\left(b\right)$implies,

g(a) = g(b)

2a=2b

a=b

Hence, g is injective.

g is surjective

Let $a\in {\square }_9$in co-domain then, g(a) = b such that 2a = b implies$\mathrm{a}=\frac{\mathrm{b}}{2}$

Thus, for any real number a in ${\square }_9$, there exists an element $\frac{b}{2}$in ${\square }_9$such that,

g (a) = 2a

= 2$\left(\frac{b}{2}\right)$

= b

Hence, g is surjective.

g is a homomorphism

Let$a,b\in {\square }_9$ then,

g(a+b) = 2(a+b)

= 2a + 2b

= g(a) + g(b)

Thus, for all $a,b\in {\square }_9$, g(a+b) = g(a) + g(b) and hence, g is a homomorphism.

Therefore, the function $g:{\square }_9\to {\square }_9$defined by$g\left(x\right)=2x$ is an isomorphism.