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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.

Thanks for reading! Have a good day :)

SumsArentDumb
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    Have you [searched our site](https://stats.stackexchange.com/search?q=group+permutation)? Other good search terms include "symmetr*" (for symmetries) and "invariant." Group theory underpins modern methods of [tag: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 https://stats.stackexchange.com/a/6135/919 for instance. Finally, at https://stats.stackexchange.com/a/288198/919 I apply the Orbit-Stabilizer theorem. – whuber Feb 22 '21 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 $S_n\times S_n$.

Tobias Windisch
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