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This is a survey article related to:normal subgroup
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Contents 1 Introduction 2 Tools in the study of distribution of normal subgroups 2.1 Lattice of normal subgroups 3 Groups with lots of normal subgroups 3.1 Abelian groups 3.2 Dedekind groups 4 Groups with very few normal subgroups 4.1 Simple groups 4.2 Quasisimple groups 4.3 Jordan-unique groups 5 Behaviour of the lattice of normal subgroups 5.1 Simple groups

Introduction

Normality is one of the more frequently encoutered group properties,, and an obvious question is: given a group, which of its subgroups are normal? What structural information about the group correlates with the way the normal subgroups are distributed within the group? This, and other related questions, are explored within this article.

Tools in the study of distribution of normal subgroups

Lattice of normal subgroups

Further information: Lattice of normal subgroups

Normality is defined as the property of being invariant under all inner automorphisms. Thus, normality is an endo-invariance property, and is hence closed under arbitrary intersections and arbitrary joins. In other words, an arbitrary intersection of normal subgroups is a normal subgroup, and the subgroup generated by any family of normal subgroups is also a normal subgroup.

For full proof, refer: Normality is intersection-closed,Normality is join-closed

Thus, the collection of normal subgroups forms a lattice with the meet operation being intersection of subgroups and the join operation being the join of subgroups (or subgroup generated). This is a sublattice of the lattice of subgroups.

Groups with lots of normal subgroups

Abelian groups

The property of being normal is an Abelian-tautological subgroup property. In other words, every subgroup of an Abelian group is normal. This follows from the fact that since elements of an Abelian group commute, the only inner automorphism of an Abelian group is the identity map. For full proof, refer: subgroup of Abelian implies normal

Note, however, that an Abelian subgroup of a non-Abelian group need not be normal. This is because inner automorphisms by elements outside that subgroup may be nontrivial.

Dedekind groups

A group in which every subgroup is normal is termed a Dedekind group or a Hamiltonian group. Interestingly there can be Dedekind groups which are not Abelian. The simplest example is the quaternion group. Moreover, all non-Abelian Dedekind groups are closely related to the quaternion group. Fill this in later

Groups with very few normal subgroups

Simple groups

In every group, the trivial subgroup and the whole group are normal. A nontrivial group for which there are no other normal subgroups, is termed a simple group. Simple groups are opposite to Dedekind groups, and the only possibility for a simple Dedekind group is the cyclic group of prime order.

Quasisimple groups

A quasisimple group is a perfect group whose inner automorphism group is simple. In a quasisimple group, the only normal subgroups are the whole group and subgroups of the center.

Jordan-unique groups

A Jordan-unique group is a group for which there is a unique composition series. In other words, there is a unique maximal normal subgroup, which in turn has a unique maximal normal subgroup, and so on.

Behaviour of the lattice of normal subgroups

Simple groups

Simple groups are the groups where the lattice of normal subgroups is as small as possible: it just has two elements