Ingleton ratio

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 ratio of the subgroup 4-tuple $$(G_1,G_2,G_3,G_4)$$ in $$G$$ is defined as:

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

The Ingleton ratio is also used to define a related notion called the Ingleton score, which is the subject of the group-theoretic formulation of the four atom conjecture.

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

$$r = \frac{|G_{14}G_{24}||G_{13}G_{23}|}{|G_1G_2||G_{34}|}$$

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

Equivalence of definitions
The equivalence between the two expressions given above for the Ingleton ratio follows from the product formula. It's easy to start from the second expression and use the product formula, then simplify to get the first expression.