He developed the concept that is today known as a normal subgroup.

Every minimal normal subgroup of a group is characteristically simple.

The socle is a direct product of minimal normal subgroups.

The center of a group is a normal subgroup.

Also, a normal subgroup of a central factor is normal.

In particular, a normal subgroup of a direct factor is normal.

It has I as normal subgroup of index 2.

A normal subgroup that is also malnormal must be one of these.

The only normal subgroup that is also abnormal is the whole group.

Simple groups have only two normal subgroups: the identity element, and M.