Exercise 1.6.11 ($A imes B \simeq B imes A$)

Question

Let $A$ and $B$ be groups. Prove that $A	imes B \simeq B 	imes A$.

Richard Ganaye · Accepted Answer

\begin{proof} Consider the map
$$
\varphi
\left\{
\begin{array}{ccl}
A 	imes B & 	o & B 	imes A\
(a,b) & \mapsto &(b,a).
\end{array}
ight.
$$
 If $\psi =  B 	imes A	o A 	imes B$ is defined by $\psi(b,a) = (a,b)$, then $\psi \circ \phi = 1_{A	imes B}$ and $\phi \circ \psi = 1_{B 	imes A}$. This proves that $\varphi$ is bijective (and $\varphi^{-1} = \psi$).
 
 Moreover, for all $(a,b) \in A 	imes B$ and all $(c,d) \in A 	imes B$,
 \begin{align*}
 \varphi [(a,b)] \varphi[(c,d)] &= (b,a)\cdot (d,c)\
 &= (bd, ac)\
 &=\varphi(ac, bd)\
 &= \varphi[(a,b)\cdot (c,d)],
 \end{align*}
 so $\varphi$ is an isomorphism, and
  $$A	imes B \simeq B 	imes A.$$
\end{proof}

Exercise 1.6.11 ($A \times B \simeq B \times A$)

Answers

Comments

Exercise 1.6.11 ($A \times B \simeq B \times A$)

Answers

Comments

Add answer