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:


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*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?
 A: There is no rationale. ${}{}{}{}{}$
A: 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."

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*https://planetmath.org/alphabetical.html#n

*https://planetmath.org/alphabetical.html#r

*https://planetmath.org/alphabetical.html#s
