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?