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

The center of a group is a normal subgroup.

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

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

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

The socle is a direct product of minimal normal subgroups.

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

The normal subgroups of the finite symmetric groups are well understood.

This normal subgroup splits the group into three cosets, shown in red, green and blue.

Some authors omit the third condition that G has no regular normal subgroup.

For a weakly c normal subgroup, we only require to be subnormal.

The torsion elements in a nilpotent group form a normal subgroup.

In fact, they are central factors, and are hence transitively normal subgroups.

Any normal subgroup is equal to its normal closure.

It is the group generated by all the normal subgroups of G that lie in H.

E.g. the congruence relations on groups correspond to the normal subgroups.

The subsets in the partition are the cosets of this normal subgroup.

All such groups have N and N as minimal normal subgroups.

Let be a normal subgroup such that .

There are no proper normal subgroups with reflections.

Galois, it is usually said, coined the word group at this time and introduced the concept of normal subgroup.

Conversely, any retract which is a normal subgroup is a direct factor.

Let G be a finite group, N be a normal subgroup.

A group is called a simple group if it does not contain a non-trivial proper normal subgroup.

All normal subgroups of G (and thus whether or not G is simple) can be recognised from its character table.