Question

Let фi : Gi ------> G1 x G2 x G3 x…….Gi x ……Gr be given by фi (gi)= (e1 ,e2 ,…..gi,…..er) where gi єGi and ej is the identity of Gj. Prove that this is a injective map.

Answer 1

Question 1.

Let $\phi_{i}:G_{i}\to G_{1}\times G_{2}\times G_{3}\times\cdots\times G_{i}\times\cdots\times G_{r}$ be given by $\phi_{i}(g_{i})=(e_{1},e_{2},\ldots,g_{i},\ldots,e_{r})$ where $g_{i}\in G_{i}$ and $e_{j}$ is the identity of $G_{j}$. Prove that this is an injective map.

Solution. Suppose $\phi_{i}(g_{i})=\phi_{i}(h_{i})$ for some $g_{i},h_{i}\in G_{i}$. By definition of $\phi_{i}$ this means that

$(e_{1},e_{2},\ldots,g_{i},\ldots,e_{r})=(e_{1},e_{2},\ldots,h_{i},\ldots,e_{r}),$

i. e.

$e_{1}=e_{1},$

$e_{2}=e_{2},$

$\ldots$

$g_{i}=h_{i},$

$e_{r}=e_{r}.$

In particular, $g_{i}=h_{i}$. Thus, $\phi_{i}(g_{i})=\phi_{i}(h_{i})$ implies $g_{i}=h_{i}$, and so $\phi_{i}$ is injective, $i=1,\ldots,r$. $\Box$