**Question.** Is it true that each uncountable group $G$ contains an uncountable subgroup $A$ and an infinite subgroup $B$ such that $A\cap B=\{1\}$? What will be the answer if we additionally require that $ab=ba$ for all $a\in A$ and $b\in B$?



Hereditarily indecomposable groups
This is not rue, by Shelah's construction of a Jonsson group
of cardinality $\aleph\_1$ (a group of size $\aleph\_1$ for which every proper subgroup is countable). See [On a problem of Kurosh, Jónsson groups, and applications](https://www.sciencedirect.com/science/article/pii/S0049237X08713466)


