Ingleton score

Definition

Suppose $G$ is a finite group and $G_{1},G_{2},G_{3},G_{4}$ are all subgroups (possibly equal, possibly distinct) of $G$. For any subset $\alpha$ of $\{1,2,3,4\}$, denote by $G_{\alpha }$ the subgroup $\bigcap _{i\in \alpha }G_{i}$. For convenience, we will write $\alpha$ simply as a concatenated string of its elements, so for instance, $G_{134}$ stands for $G_{\{1,3,4\}}$ and is defined as $G_{1}\cap G_{3}\cap G_{4}$.

The Ingleton score $s$ of this tuple is defined as follows, where the base of logarithms is chosen to be the same for the numerator and the denominator:

$s={\frac {\log r}{\log |G/G_{1234}|}}$

where $r$, also called the Ingleton ratio, is defined as:

$r={\frac {|G_{12}||G_{13}||G_{14}||G_{23}||G_{24}|}{|G_{1}||G_{2}||G_{34}||G_{123}||G_{124}|}}$

Using the product formula, the Ingleton ratio can be rewritten as:

$r={\frac {|G_{14}G_{24}||G_{13}G_{23}|}{|G_{1}G_{2}||G_{34}|}}$

Note that the sets whose orders are being taken here are products of subgroups, but need not be subgroups themselves.

Facts

• For obvious reasons, the Ingleton score is at most 1. Further information: Ingleton score is at most one
• The four atom conjecture states that the Ingleton score is at most a certain number whose decimal approximation reads $0.089373$.