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.

One of the many basic questions in group theory is the deep similarity as well as contrast in the definitions of normal and characteristic subgroup: a priori, the notion of characteristic subgroup (as being a subgroup invariant under all automorphisms) seems more natural than the notion of normal subgroup (as being a subgroup invariant under only the inner automorphisms). However, as we shall see here, normal subgroups are more truly characteristic since they capture invariance under automorphisms that remain automorphism even after putting in a lot of additional structure. Further, we shall also see that normal subgroups, on account of being both subgroups and ideals, are a very powerful tool in group theory.

This article looks at the many reasons why normality keeps popping up at odd places 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:

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 particular, 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 by 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.

Subgroups of the automorphism group of a topological space

Given any topological space, we can consider its self-homeomorphism group: the automorphism group in the category of topological spaces with continuous maps. We can now consider subgroups of this group, and any subgroup that we can pick uniquely must be a normal subgroup. That's because any homeomorphism acts on the self-homeomorphism group itself by conjugation, and hence any subgroup determined uniquely must be normal.

Thus, for instance, the subgroup of self-homeomorphisms isotopic to the identity is a normal subgroup.

Subgroups of the automorphism group of a group

Again, any subgroup of the automorphism group of a group G that can be picked uniquely given G, must be invariant under automorphisms of G, which become inner automorphisms in Aut(G). Thus,for instance, the group of class-preserving automorphisms, normal automorphisms, inner automorphisms, and many others, are normal subgroups in Aut(G).

In topological groups

Inner automorphisms are monomial

Inner automorphisms of a group are special in the sense that they can be expressed directly in terms of the group operations. In other words, we can actually write an expression in terms of group elements that can be used to compute the image of an element under an inner automorphism.

In general, an automorphism for which we can provide an algebraic formula is termed a monomial automorphism. Thus, all inner automorphisms are monomial.

All monomial automorphisms are continuous

A topological group is a set equipped with both the structure of a group and the structure of a topological space, such that all the group operations are continuous with respect to the topological space structure. A topological automorphism of a topological group is defined as a bijection from the topological group to itself that is group-theoretically an automorphism, and also a homeomorphism of the topological space structure. Clearly, all group-theoretic automorphisms of a topological group need not be continuous.

However, what is true is that every monomial automorphism of a topological group is continuous. In particular, every inner automorphism is a homeomorphism (since both that and its inverse are continuous) and is hence a topological automorphism.

Topological subgroup-defining functions

We had earlier looked at two notions: subgroup-defining functions in the context of an abstract group, and functions that define subgroups in the context of a particular group action. We can now look at subgroup-defining functions in the context of the additional topological structure. These use both the group structure and the topological space structure to pinpoint a subgroup.

For instance, we may be looking at the connected component containing the identity, or the path-connected component containing the identity, or the quasicomponent containing the identity.

Clearly, just like any subgroup-defining function always yields a characteristic subgroup, any topological subgroup-defining function always yields a subgroup that is invariant under topological automorphisms (that is, automorphisms that are automorphisms of the group structure as well as homeomorphisms topologically). Further, since any inner automorphism is topological, every topological subgroup-defining function must yield a normal (though not necessarily a characteristic) subgroup.

In particular:

  • The connected component of the identity in a topological group is a normal subgroup
  • The path-connected component of the identity in a topological group is a normal subgroup

Similar phenomena for other additional structures

Instead of a topological structure, we may equip the group with additional analytic structure (hence turning it into a Lie group), algebraic structure (hence turning it into an algebraic group), or some other suitable structure. For each such notion of structure, there are corresponding restricted versions of continuity (analyticity in the case of Lie groups, regularity in the case of algebraic groups) for which we assume that the multiplication and inversion maps are continuous. This greater generality of a group with additional structure is also dubbed as a group scheme and each particular element of that is termed a group object.

For any group scheme, any monomial map is continuous, since it involves composing the group operations. Hence, in particular, inner automorphisms are automorphisms even at the level of the group scheme (since both they and their inverses are continuous). Thus, any subgroup-defining function in the context of the additional scheme structure, must still yield a normal subgroup (though the subgroup yielded may be far from characteristic).

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 \forall a,b,c \in G

a * e = e * a = a \forall a \in G

a * a^{-1} = a^{-1} * a = e \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 are, in the general theory of universal algebra, termed kernels.

There is a notion of ideal: An ideal I of an algebra A in a variety of algebras is a subset such that for any formula f(x_1,x_2,\ldots,x_r,y_1,\ldots,y_s) which always takes the value 0 when all x_is are 0, the formula always yields an element in the subset when all the x_is are in I and y_js are in A.

Every kernel in an algebra with zero is an ideal. In the variety of groups, two remarkable things happen:

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. Further information: ideals are subalgebras in the variety of groups

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.

Another remarkable aspect

Further information: characteristic subalgebras are ideals in the variety of groups

In the opposite direction, the remarkable thing is that whenever we have a situation where we are guaranteed to obtain a normal subgroup, we can take a quotient group. Thus, all the situations described here, which include subgroup-defining functions, topological subgroup-defining functions, and uniquely picked out subgroups in automorphism groups, are normal, and we can take the associated quotient groups.

Normal field extensions and normal subgroups

Normal extensions

Historically, the concept of normal subgroup arose from Galois's study of field extensions. Galois was considering algebraic field exntesions: a field K containing a base field k such that every element of K is algebraic (that is, satisfies a polynomial) over k.

Galois called a separable algebraic extension normal if whenever a polynomial over k has one root in K, all its roots are in K.

The Galois correspondence

Galois proved the following Fundamental Theorem of Galois Theory. Let K/k be an extension of fields, and G = Aut(K/k) (that is, the field automorphisms of K that fix k pointwise).

Then, consider the following relation between g \in G and \alpha \in K: g and \alpha are related if g.\alpha = \alpha.

Under this relation we get two maps:

  • A map that takes as input a subgroup H of G and outputs the subfield of K comprising those elements that are invariant under every element of H.
  • A map that takes as input a subfield L of K and outputs the subgroup of G comprising those elements that fix every element of L

The Fundamental Theorem of Galois Theory states that these two maps are inverses of each other. In other words, every subgroup and every subfield is closed under the Galois correspondence.

The second part of the Fundamental Theorem states that under this correspondence, the normal subgroups of G get identified with those subfields L that are normal field extensions over k.

Tihs can be used as a definition of normality for any group that arises as the automorphism group of some nrmal separable algebraic field extension.