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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.

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    $\begingroup$ Have you searched our site? Other good search terms include "symmetr*" (for symmetries) and "invariant." Group theory underpins modern methods of combinatorics. Some more advanced abstract algebra methods (still at the undergraduate level), such as rings and ideals, have been exploited here to solve some problems: see stats.stackexchange.com/a/6135/919 for instance. Finally, at stats.stackexchange.com/a/288198/919 I apply the Orbit-Stabilizer theorem. $\endgroup$
    – whuber ♦
    Commented Feb 22, 2021 at 20:36

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The Lecture on algebraic statistics provides an excellent introduction into this topic. A more comprehensive overview is provided by Seth Sullivants book Algebraic statistics.

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 $S_n\times S_n$.

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