Let G be a group and be a subgroup of G such that H notequal

Let G be a group and be a subgroup of G such that H notequalto G. It is said that His a maximal subgroup of G if there is no subgroup F of G such that H F and F notequalto H, G. Similarly, H is said to be a maximal normal subgroup of G if H G and there is no normal subgroup F of G such that H F and F notequalto H, G. Is it true that any group has a maximal normal subgroup?

Solution

It is not true that every group has a maximal normal subgroup. It is because an infinite abelin group does not have any such subgroup.