a group whose only normal subgroups are itself and the identity

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 the subgroup comprising only the identity and the group itself (the so called trivial subgroups)

a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group

In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself.