# Are permutation groups used in statistics?

I was told that there may be a connection between concepts in abstract algebra and applications in statistics (also confirmed by the 'algebraic statistics' tag on this website). I had no idea, so I'm curious. In particular, I wanted to understand how might permutation groups be used, as I believe they are a central and interesting concept in abstract algebra.

I'm trying to look into the theory online, but all I see are links to genuine research articles or other unhelpful things for me... I'm not a professional... If you can give me some examples, or perhaps easy to understand resources on such a topic, I would be glad.

I have an undergraduate understanding in statistics and algebra, so try not to use statistical word wizardry on this small tadpole of a mathematician.

One key player of algebraic statistics are toric ideals $$I_A$$ motivated by a matrix $$A$$ coming from a log-linear model. Typically, one wants to find a finite generating set of $$I_A$$ (called a Markov basis) as they can be used for statistical tests (like Fisher's exact test, you may also want to have a look here). Groups may appear in giving descriptions of the generating sets of those ideals, as typically, symmetry on the variables plays an essential role.
In section 4 of the seminal paper by Persi Diaconis and Bernd Sturmfels, an example of such a generating set of the toric ideal (here, even an universal Gröbner basis) of the "no-three-factor interaction" model using $$S_n\times S_n$$.