I am trying to solve a problem which says,Let $G$ be a group and $H$ be a subgroup of finite index.show that $G$ contains a normal subgroup of finite index.
Actually I cann't except this statement as I know if we consider $A_5$ I get a cyclic subgroup generated by $(1 2)(1 3)(1 4)(1 5)$,Then we get a subgroup with index $14$.According to the above statement $A_5$ has a normal subgroup of finite index which contradicts to the fact that $A_5$ is simple.
My question is am I correct to read the question or I have done some silly mistake to understand it.If I am wrong then how should I proceed,give me a hint in that case,please.