# What is it to be normal?

I'm interested to find out why the word 'normal' crops up so many different areas of mathematics, especially but not exclusively in abstract algebra, and how the definitions are related, if at all.

Some common examples are:

• A subgroup $H$ of a group $G$ is normal if $gHg^{-1}=H$ for each $g \in G$.

• An algebraic extension $L$ of a field $K$ is normal if every polynomial in $K[X]$ with a root in $L$ splits in $L$.

• A topological space $X$ is normal if for any disjoint closed subsets $A,B \subseteq X$ there exist disjoint open subsets $U,V \subseteq X$ with $A \subseteq U$ and $B \subseteq V$.

• A real number is normal if, in each base $b$, each of the digits from $0$ to $b-1$ has asymptotic density $\frac{1}{b}$ in its base-$b$ expansion.

• A vector $v \in \mathbb{R}^3$ is normal to a $2$-manifold $X$ at the point $p \in X$ if $\langle v, w \rangle = 0$ for each $w \in T_p X$.

• A random variable $X : (\Omega, \mathcal{F}, \mathbb{P}) \to \mathbb{R}$ is normal if its probability density function takes the form $\dfrac{1}{\sqrt{2\pi \sigma^2}} \exp \left \{ -\dfrac{(x-\mu)^2}{2\sigma^2} \right \}$ for some $\mu \in \mathbb{R}$ and $\sigma^2 > 0$.

Normal groups and normal field extensions are related thanks to Galois theory: if $F/K$ is a Galois extension with Galois group $G$ then $H \le G$ is a normal subgroup if and only if $F^H/K$ is a normal extension. But how about normal subgroups and normal topological spaces, for example?

Is there a rationale behind using the word normal, or are the meanings disjoint, having evolved in separate fields for unrelated reasons?

Or, to whittle this all down to a single question: is there a well-defined notion of 'normality' in mathematics, and if so, what is it?

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+10 for getting the spelling of the word rationale right! :) – Mariano Suárez-Alvarez Jan 9 '12 at 20:13
I doubt you will find much commonality with the usages of the word "normal," but the right angle definition might be the oldest form, given that a dictionary tells me it comes from the Latin word "normalis," which means "made according to a carpenter's square." – Thomas Andrews Jan 9 '12 at 21:16
An interesting question is that the notion of a "normal subgroup" is one on which left and right actions are the same, so is the word "normal" used because a balance between left and right makes it "like" a right angle, or was it chosen for its more obvious "nice" meaning? Oddly, normal subgroups are not very "normal" in the probabilistic sense - at least, not in non-commutative groups. :) – Thomas Andrews Jan 9 '12 at 21:37
All of them are named after Karl-Heinz Normal. A biographical sketch of him appeared in the mathematical intelligencer in (I think?) the '80s, along with those of Victoria Cross (noted for the cross-product, the cross-ratio, her word puzzles, and her style of country running), Montmorency Royce Sebastian Carlow (you've heard of Monty Carlow methods), and a number of others. – Michael Hardy Jan 10 '12 at 0:15
I also seem to recall Charles Delauncy Branch ("Branch points"). His mother's first name was Olive. – Michael Hardy Jan 10 '12 at 1:16

If I knew the answer to the title question, I probably wouldn't spend quite so much time at MO and MSE. (ba-dum ching!)

But seriously...

As a general rule, I imagine that the term arises for the simple reason that when you start studying a type of object (subgroup, topological space, field extensions, probability distributions), you quickly stumble upon the "good" set of such objects -- the ones that behave the way you want them to behave in order to set up a general theory. You then call these the "normal" such objects (because "good" sounds a little silly? But then you find "excellent" rings...) and build your theory from there up. Of course, a lot of the uses of the word are related to each other (as you indicate in a comment, there's a link between normal field extensions and normal subgroups), and certainly some uses of the word normal come from other sources. For example, the use of norms and normal vectors in linear algebra, according to the OED, probably date back to the 17th century use of the norm to reference right angles.

In any case, support for this explanation comes from the likewise extraordinary number of uses of terms related to "normal" (e.g., "simple", "regular", etc.), which are natural adjective to ascribe to basic objects if you're starting a theory from the ground up. It's entertaining to check out the sheer length of the PlanetMath encyclopedia entries beginning with N, R, and S due to the preponderance of terms that start with "normal", "regular," and "simple."

http://planetmath.org/encyclopedia/N/

http://planetmath.org/encyclopedia/R/

http://planetmath.org/encyclopedia/S/

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– Cam McLeman Jan 9 '12 at 21:09
A funny exception of using the word normal for the good objects is Jordan algebras. One classifies them in two groups: they special ones and the exceptional ones :) – Mariano Suárez-Alvarez Jan 10 '12 at 1:36
"Singular" comes to mind as well. – Cam McLeman Jan 20 '12 at 16:16

There is no rationale. ${}{}{}{}{}$

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I thought that might be the case. But there is some relationship between the term's use within, say, abstract algebra. Is it more likely that a 'normal extension' was so called because of its relationship with normal subgroups, or is there an idea of 'normal' in algebraic contexts which would apply to both? The question, I realise, is quite naïve, but I'm interested. – Clive Newstead Jan 9 '12 at 20:31