Characteristic subgroup of finite group

Definition
A subgroup of a group is termed a characteristic subgroup of finite group if it satisfies the following equivalent conditions:


 * 1) The whole group is a finite group and the subgroup is a characteristic subgroup of it.
 * 2) The whole group is a finite group and the subgroup is a conjunction involving::strictly characteristic subgroup (i.e., invariant under all surjective endomorphisms) of it.
 * 3) The whole group is a finite group and the subgroup is an conjunction involving::injective endomorphism-invariant subgroup  of it.