Question

If$f:G_1\to H_1$ and$g:G_2\to H_2$ are isomorphisms of groups, prove that the map$\theta :G_1\times G_2\to H_1\times H_2$ given by$\theta \left(a,b\right)=\left(f\left(a\right),g\left(b\right)\right)$ is an isomorphism.

Short answer

It is proved that, the map $\theta :G_1\times G_2\to H_1\times H_2$is an isomorphism.

Step-by-step solution

Isomorphic Groups

Two groups G and H are said to be isomorphic if there exists an isomorphism f fromG onto H .

Prove that the map θ : G1×G2→H1×H2 given by θa,b=fa,gb is an isomorphism.

To show isomorphism, just show mapping is one-one, onto, and homomorphism.

Check for one-one.

Let$\theta \left(a,b\right)=\theta \left(c,d\right)$for $a,c\in G_1$and $b,d\in G_2$.

This implies that$f\left(a\right)=f\left(c\right)$and $g\left(b\right)=g\left(d\right)$.

Since f and g are given to be isomorphism,$a=c$and$b=d$.

Therefore, $\left(a,b\right)=\left(c,d\right)$.

Hence, the map$\theta :G_1\times G_2\to H_1\times H_2$is one-one.

Check for onto as:

Let $\left(c,d\right)\in H_1\times H_2$.

This implies that $c\in H_1$and $d\in H_2$.

Since f and g are given to be isomorphism, there existsrole="math" localid="1653577877953" $a\in G_1$and$b\in G_2$such that $f\left(a\right)=c$and$g\left(b\right)=d$.

This implies that there exists $\left(a,b\right)\in G_1\times G_2$such that:

$\begin{array}{rcl}\theta \left(a,b\right)& =& \left(f\left(a\right),g\left(b\right)\right)\\ & =& \left(c,d\right)\end{array}$

Hence, the map$\theta :G_1\times G_2\to H_1\times H_2$ is onto.

Check for homomorphism as:

Show that$\theta \left(\alpha \left(a,b\right)+\left(c,d\right)\right)=\alpha \theta \left(a,b\right)+\theta \left(c,d\right)$ as:

role="math" localid="1653578341194" $\begin{array}{rcl}\theta \left(\alpha \left(a,b\right)+\left(c,d\right)\right)& =& \theta \left(\left(\alpha a,\alpha b\right)+\left(c,d\right)\right)\\ & =& \theta \left(\alpha a+c,\alpha b+d\right)\\ & =& \left(f\left(\alpha a+c\right),g\left(\alpha b+d\right)\right)\end{array}$

Since f and g are given to be isomorphism, further simplify as:

$\begin{array}{rcl}\theta \left(\alpha \left(a,b\right)+\left(c,d\right)\right)& =& \left(\alpha f\left(a\right)+f\left(c\right),\alpha g\left(b\right)+g\left(d\right)\right)\\ & =& \left(\alpha f\left(a\right),\alpha g\left(b\right)\right)+\left(f\left(c\right),g\left(d\right)\right)\\ & =& \alpha \left(f\left(a\right),g\left(b\right)\right)+\left(f\left(c\right),g\left(d\right)\right)\\ & =& \alpha \theta \left(a,b\right)+\theta \left(c,d\right)\end{array}$

Thus, the map$\theta :G_1\times G_2\to H_1\times H_2$ is homomorphism.

Therefore, the map $\theta :G_1\times G_2\to H_1\times H_2$is an isomorphism.