For a subgroup H $H$ of a given group G, $G$, I say H $H$ is "big" if it has nonempty intersection with each conjugacy class of G. $G$. I have known that, trivially, G $G$ itself is "big". And if H $H$ is a normal subgroup and it is "big", then H=G. $H=G$. I have also known that a finite group has no proper "big" subgroup. My question is "Is there an infinite group who has a proper 'big' subgroup?"