I've been struggling with this problem for some time now so any help would be greatly appreciated.
I have a finite group $G$ and a subset $A$ of $G$. I am told that $G$ acts transitively on $A$ and that $A$ generates $G$.
Furthermore I have two subgroups of $G$ called $U_1$ and $U_2$ such that $A$ is contained in $U_1\cup U_2$. I now have to prove that either $U_1=G$ or $U_2=G$.
What I've done so far:
I've tried solving it by contradiction so I assume that $U_1$ and $U_2$ are different from $G$. I found it fairly easy to say a few things about $U_1$ and $U_2$ such that they aren't equal, one is not contained in the other and their union isn't $G$ but I don't know what to do about the group action part.