A group is simple if it has no proper nontrivial normal subgroups. An abelian finite simple group has to be cyclic of prime order. A nonabelian finite simple group must have even order, by the Feit-Thompson theorem.

a group whose only normal subgroups are itself and the identity

a group whose only normal subgroups are the subgroup comprising only the identity and the group itself (the so called trivial subgroups)