Exercise 1.6.12 ($(A imes B) imes C \simeq A imes (B imes C)$)

Question

Let $A, B$ and $C$ be groups and let $G = A	imes B$ and $H = B 	imes C$. Prove that $G 	imes C \simeq A 	imes H$.

Richard Ganaye · Accepted Answer

\begin{proof}

Consider the map
$$
\varphi
\left\{
\begin{array}{ccl}
G 	imes C & 	o &A 	imes H\
((a,b), c) & \mapsto & (a, (b,c))
\end{array}
ight.
$$
This map is well defined, because every element  $g \in G$ can be written uniquely under the form $g = (a,b)$ where $a \in A, b \in B$. Then 
$(a, (b,c)) \in A 	imes H$.

We define $\psi$ by
$$
\psi
\left\{
\begin{array}{ccl}
A 	imes H & 	o &G 	imes C \
(a, (b,c)) & \mapsto & ((a,b), c)
\end{array}
ight.
$$
Similarly, $\psi$ is well defined, and $\psi \circ \varphi = 1_{G	imes C},\ \varphi 	imes \psi = 1_{A 	imes H}$. Therefore $\varphi$ is bijective, and $\varphi^{-1} = \psi$.

Moreover, if $((a, b),c))$ and $((d,e), f)$ are in $G 	imes C $, then
\begin{align*}
\varphi[((a, b),c)] \varphi[((d,e), f)] &= (a,(b,c)) \cdot (d,(e,f))\
&= (ad, (b,c) \cdot (e,f))\
&=(ad, (be,cf))\
&= \varphi[((ad, be), cf)]\
&=\varphi[((a,b)\cdot (d,e)),cf)]\
&=\varphi[((a,b) ,c) \cdot ((d,e),f)].
\end{align*}

This shows that $\varphi$ is an isomorphism, so
$$G 	imes C \simeq A 	imes H.$$

\bigskip
Since $G 	imes C =  (A 	imes B) 	imes C$ and $A 	imes H = A 	imes (B 	imes C)$, we have proved $$(A 	imes B) 	imes C \simeq  A 	imes (B 	imes C).$$
\end{proof}

Exercise 1.6.12 ($(A \times B) \times C \simeq A \times (B \times C)$)

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Exercise 1.6.12 ($(A \times B) \times C \simeq A \times (B \times C)$)

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