Variational formulations in group theory?

Question

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = \sup\{\phi(g) : g \in G\}, $$ where $\alpha$ is a number (or set) determined by some property of the group $G$ and $\phi$ is a real-valued (or set-valued) function on $G$. For a silly example, take $\phi(g)$ to be the order of an element and $\alpha$ the order of the group; the group is cyclic iff $\alpha(G) = \#G = \sup_{g \in G}\phi(g).$

I would also be interested in results on the probability distributions induced on groups or reflected in their structure in some way. For instance, suppose a finite group $G$ acts on a finite set $A$, and let $\phi : G \rightarrow \mathbb{N}_{\geq 1}$ be a function such that $\phi(g) = \#\{a \in A : g\cdot a = a\}$. In other words, $\phi$ assigns to each $g \in G$ the number of fixed points it has in $A$.

Now let $X$ be a random variable uniformly distributed on the finite set $G$. Then Burnside's lemma states that the expectation $\mathbb{E}[\phi(X)]$ gives us the number of orbits of the action: $$|A/G| = \mathbb{E}[\phi(X)] \stackrel{\cdot}{=} \sum_{g \in G} \frac{\phi(g)}{|G|}.$$ I'd appreciate any key words / things to Google. I assume many things come from number theory.

Answer 1

There are several strong theorems about random variables on groups, that might be something you were looking for:

Generation Formula:

Suppose $G$ is a finite group, $\{X_i\}_{i = 1}^{n}$ are i.i.d uniformly distributed random elements of $G$. Then $P(\langle \{X_i\}_{i = 1}^{n} \rangle = G) = \sum_{H \leq G} \mu(G, H) {\left(\frac{|H|}{|G|}\right)}^n$, where $\mu$ is the Moebius function for subgroup lattice of $G$.

Generalized Erdos-Turan theorem:

Suppose $G$ is a finite group. $\{X_i\}_{i = 1}^{n}$ are i.i.d uniformly distributed random elements of $G$. Then $G$ is nilpotent of class $n$ iff $P([ … [[a_0, a_1], a_2]… a_n] = e) > 1 - \frac{3}{2^{n + 2}}$

Law of large numbers for groups:

Suppose $G$ is a finitely generated group, $A$ is a finite set of generators of $G$. $d: G\times G \to \mathbb{N}$, is the metric on $G$ induced by the Cayley Graph $Cay(G, A)$. Suppose $\{X_i\}_{i = 1}^\infty$ is a sequence of i.i.d. random elements of $G$, such that $E(d(X_i, e)) < \infty$, and $Z_n = \Pi_{i=1}^n X_i$. Then $\exists \alpha \in \mathbb{R}, P(\lim_{n \to \infty} \frac{d(Z_n, e)}{n} = \alpha) = 1$.