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.

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