Chapter 3.5, Problem 15E

Consider the additive group $ℝ$ of real numbers. Prove or disprove that each of the following mappings $\varnothing :ℝ\times ℝ\to ℝ$ is an isomorphism.

a. $\varnothing (x,y)=x$

b. $\varnothing (x,y)=x+y$

Sec. $3.1,\#52\gg$

Let $G_1$ and $G_2$ be groups with respect to addition. Define equality and addition in the Cartesian product by $G_1\times G_2$

$(a,b)=(a^{\prime },b^{\prime })$ if and only if $a=a^{\prime }$

and $b=a^{\prime }$

$(a,b)+(c,d)=(a\oplus c,b⊞d)$

Where $\oplus$ indicates the addition in $G_1$ and $⊞$ indicates the addition in $G_2$.

Prove that $G_1\times G_2$ is a group with respect to addition.

Prove that $G_1\times G_2$ is abelian if both $G_1$ and $G_2$ are abelian.

For notational simplicity, write

$(a,b)+(c,d)=(a+c,b+d)$

As long as it is understood that the additions in $G_1$ and $G_2$ may not be the same binary operations.