Ubiquity of normality

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Introduction

Normality is one of the most important subgroup properties, with a long and chequered history as well as a knack of appearing almost as ubiquitously as groups themselves. In this article, we look at the many reasons why normality is an important subgroup property, and why it keeps popping up repeatedly.

This article looks at the many reasons why normality keeps popping up at odd plcaes wherever groups do.

Subgroup-defining functions yield normal subgroups

What is a subgroup-defining function?

Further information: subgroup-defining function

A subgroup-defining function is a function that takes in a group and outputs a unique subgroup of that group. For instance:

• The center is the set of those elements that commute with every element
• The commutator subgroup is the subgroup generated by the commutators
• The Frattini subgroup is the intersection of all maximal subgroups

All subgroup-defining functions yield characteristic subgroups

Subgroup-defining functions satisfy the condition of being invariant under isomorphism: any isomorphism of groups, preserves the function. Hence, in paritcular, the subgroup must be invariant under all automorphisms, and hence, must be a characteristic subgroup.

Now, the condition of being a characteristic subgroup requires invariance under all automorphisms. In particular, it is true that every characteristic subgroup is invariant under all inner automorphisms, that is, all automorphisms described via conjugation by a group element. Hence, every subgroup-defining function yields a normal subgroup.

Thus, the center, commutator subgroup, Frattini subgroup etc. are all characteristic, and hence, normal subgroups.

Invariance under own action

Consider the automorphism group of a structure. Then, the automorphism group acts on the structure byb relabelling the structure (for instance, when a group acts on a set, it relabels the elements of the set). Hence, the automorphism group acts on itself by conjugation, and this action can be viewed in terms of just changing the labels on the underlying set.

We may be able to define subgroups of the automorphism group of the structure, in the context of its action. That is, the definitions we give now are not subgroup-defining functions, rather they are functions that depend, for their definition, on the way the group acts on the structure. The subgroups thus obtained need not be characteristic.

However, it continues to be true that any subgroup thus obtained must be a normal subgroup, because any such subgroup must be invariant under a relabelling of the underlying set, and hence, must be invariant under inner automorphisms of the whole automorphism group.

Thus, for instance, the group of class automorphisms, extensible automorphisms and many others, are all normal subgroups of the whole automorphism group of a group.

Normal subgroups as ideals

We know the following fact: normal subgroups are precisely the kernels of homomorphisms. Thus, any place where we are interested in the study of quotients of groups, normal subgroups pop in automatically as the kernels.

To understand the statement and its deeper implications, let us look at the more general context of homomorphisms in a variety of algebras.

Variety of algebras

In the theory of universal algebra, a variety of algebras is a collection of algebras (each with a marked collection of operations) that is closed under taking subalgebras, quotients and arbitrary direct products.

Every variety of algebras is equational, that is, an algebra with those operations belongs to the variety if and only if it satisfies some system of identities with all the variables universally quantified.

For instance, the variety of groups is described by three operations:

• The constant operation that produces the identity element
• The unary operation that takes an element and outputs its inverse
• The binary operation that takes two elements and outputs their product

Subject to the following three laws:

• The associativity of the binary operation (multiplication)
• The fact that the identity element is a multiplicative identity
• The fact that the inverse operation gives the inverse with respect to the binary operation

All these laws can be stated as universally satisfied identities. If $G$ is the underlying set, $*$ denotes the group multiplication, $g^{-1}$ denots the inverse of $g$ and $e$ denotes the multiplicative identity for group multiplication, then:

$a*(b*c)=(a*b)*c\mathrm{\forall }a,b,c\in G$

$a*e=e*a=a\mathrm{\forall }a\in G$

$a*a^{-1}=a^{-1}*a=e\mathrm{\forall }a\in G$

Similarly, the commutative rings with identity form a variety of algebras. Here, there are five operations:

• The binary operations $+$ (addition) and $*$ (multiplication)
• The unary prefix operation $-$ takes as element and outputs the additive inverse
• The constant operations $0$ and $1$ output the additive and multiplicative identities respectively

Variety of algebras with zero

A variety of algebras with zero is a variety of algebras with a special constnat operation, called the zero operation.

For instance, the variety of groups can be viewed as a variety of algebras with zero, where the zero is the identity element. The variety of rings can be viewed as a variety of algebras with zero, where the zero is the additive identity element.

Congruence on an algebra

Given two algebras $A$ and $B$ of the same variety, a homomorphism from $A$ to $B$ is a map that commutes with all the algebra operations. This reduces to the usual notion of homomorphism when we deal with the variety of groups, the variety of commutative rings with identity, and so on.

A congruence on an algebra is an equivalence relation such that the algebra structure descends naturally to the quotient. In other words, a congruence of an equivalence relation that arises as the relation of being in ther same fibre of a surjective homomorphism of algebras.

For instance, when we are looking at the variety of groups, the congruence classes of a congruence are the cosets of a normal subgroup (in other words, every congruence arises from a normal subgroup, as the equivalence relation of being in the cosets).

Similarly, for the variety of commutative rings with identity, the congruence classes of a congruence are the cosets of an ideal.

Congruence on an algebra with zero

Given a variety with zero, and a congruence on an algebra in the variety, we can look at a particular congruence class for the congruence -- namely, that congruence class that arises as the inverse image of zero under the congruence.

Such congruence classes, in the general context of universal algebra, are termed ideals.

What we have essentially said thus is that:

In the variety of groups (with zero viewed as the identity element) the ideals are precisely the normal subgroups.

What makes this remarkable

Ideals are definitely important as they arise naturally in the study of quotient maps, and hence normal subgroups are naturally important. What makes the theory of groups remarkable, however, is the fact that normal subgroups are both ideals and subalgebras.

This is not true in most varieties -- for instance, in the variety of commutative rings with identity, the ideals are not subrings (as they do not contain the multiplicative identity).

Thus, it is only in groups that we can make sense of an ideal (read: normal subgroup) of an ideal (read: normal subgroup), because the ideal itself is a group. This enables us to talk of normal series, composition series and the like.