For groups $A,B,C$, if $A\times B$ and $A\times C$ are isomorphic do we have $B$ isomorphic to $C$? [duplicate]

Solution 1:

It is difficult to determine whether or not a group $G$ is cancellable, i.e. $H \times G \simeq K \times G$ implies $H \simeq K$ for every groups $H,K$. Hirshon proved that finite groups are cancellable. Also, Vipul Naik gave a proof I find very nice of the fact that $A \times B \simeq A \times C$ implies $B \simeq C$ for every finite groups $A,B,C$. This property holds also when $A,B,C$ are finitely generated abelian groups or divisible groups; however, in his book Infinite abelian groups, Kaplansky claims that the problem is open when $A,B,C$ are arbitrary abelian groups.

A counterexample exists when $A,B,C$ are finitely presented. Let $$\Pi = \langle a,b,c \mid [a,b]=[a,c]=1, \ b^{11}=1, \ cbc^{-1}=b^4 \rangle,$$ $$H= \langle b,c \mid b^{11}=1, \ cbc^{-1}b^4 \rangle,$$ and $$K=\langle b,y \mid b^{11} , \ yby^{-1}=b^5 \rangle.$$ It is not difficult to notice that $\Pi\simeq\mathbb{Z}\times H$. With some supplementary work, it can be proved that $\Pi \simeq \mathbb{Z} \times K$, hence $$\mathbb{Z} \times H \simeq \Pi \simeq \mathbb{Z} \times K.$$ However, $H$ and $K$ are not isomorphic. See here for more information.

Therefore, $\mathbb{Z}$ is not a cancellable group, even in the class of finitely presented groups.

In fact, Francis Oger linked several cancellation problems to model theory of groups, for example: If $G$ and $H$ are finitely generated finite-by-nilpotent groups, then $G \times \mathbb{Z} \simeq H \times \mathbb{Z}$ iff $G$ and $H$ are elementary equivalent. See here and the references therein.