For a finite group $G$ and its subgroup $H$. Show that there is a subset $T$ which is simultaneously a transversal for the left and right cosets of $H$.
Further ,if we consider any two partition of $G$,and the sets have the same cardinality.Can we find a transversal for both of them?
I get the answer as the comments tell me how to do it.Considerating the left cosets as point in left,and right cosets in right.Two points are adjacent when they are in different sides and have common elements.Then I realized that if there exists a complement match,then I can prove it.The condition of Hall's theorem is easy to be seen as it will lead to a controdiction of the cardinality as they are partitions with the sets of same cardinality.
However,when it comes to the sitiation of infinte cardinality of G,what can we say about it?