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:

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

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

$r=\frac{\left|G_{12}\right|\left|G_{13}\right|\left|G_{14}\right|\left|G_{23}\right|\left|G_{24}\right|}{\left|G_1\right|\left|G_2\right|\left|G_{34}\right|\left|G_{123}\right|\left|G_{124}\right|}$

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

$r=\frac{\left|G_{14}{G}_{24}\right|\left|G_{13}{G}_{23}\right|}{\left|G_1{G}_2\right|\left|G_{34}\right|}$

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, and it is not hard to see that this inequality is strict. 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 $0.089373$.