Invitation to finite groups
Teo Banica
TL;DR
This work assembles a broad introduction to finite groups, beginning with core group-theoretic foundations and concrete examples such as $\mathbb Z_N$, $D_N$, and $S_N$, and progressing to representations through permutation matrices and linear groups $O_N$ and $U_N$. It develops the structure of cyclic and finite abelian groups, the role of homomorphisms and quotients, and the geometry of dihedral and symmetric groups, along with detailed treatment of normality and element orders. The text then shifts to permutation theory, via Cayley’s theorem, permutation representations, and cycle/signature decompositions, tying these to graph symmetries and circulant/transitive graphs, and setting the stage for deeper representation-theoretic topics such as Peter–Weyl, tensor categories, Tannaka duality, and quantum groups. Overall, it builds a foundation for understanding finite groups, their representations, and broad connections to geometry, topology, and probability.
Abstract
This is an introduction to the finite groups, with focus on the groups of permutations and reflections, and more generally, on the finite groups of unitary matrices. We first discuss the basics of group theory, featuring the cyclic, dihedral and symmetric groups, and the structure result for finite abelian groups. Then we study the complex reflection groups, with general theory and examples, classification results, and with a look into braid groups too. We then go into the study of representation theory, and of more advanced aspects, such as Tannakian duality, Brauer theorems and Clebsch-Gordan rules. Finally, we discuss, using representation theory methods, a number of advanced analytic aspects, for the most in relation with questions coming from probability.