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Problems for seminar 1 of the course "Algebra 1" (A. A. Avilov, Independent University of Moscow, autumn 2026), 7 September 2026 — groups, normal subgroups and quotients, the symmetric group. Classification of groups of order 3, 4, 6, 10; groups in which every element has order 2; generators and relations for $D_n$; finite generation of $\mathbb{Q}$ and $\mathbb{Q}^*$; the centre, subgroups and normal subgroups of $D_n$; direct sums; the Chinese remainder theorem; cycles and cycle types in $S_6$, orders of elements of $S_{10}$; generators of $S_n$; inner automorphisms; automorphism groups of $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $S_3$, $S_4$.

Finite geometries — the Platonic solids. A reminder — right actions, orbits, stabilisers, fundamental domains (corrected definition) and the example of $\mathbb{Z} + \mathbb{Z}$ on the plane; the formula $|G| = |\operatorname{Orb}(x)| \cdot |\operatorname{Stab}(x)|$ and the 48 symmetries of the cube; regular polyhedra — Sosinsky's definition and the definition via flags, a stellated counterexample; the five Platonic solids; regular polytopes in $\mathbb{R}^4$ (six) and in $\mathbb{R}^n$ for $n \ge 5$ (the cube, the simplex, the hyperoctahedron) with the coordinates of their vertices; why there are no more than five — face angles and the Schläfli symbol; uniqueness of the gluing and the rigidity theorem; Euler's formula; the finite subgroups of $SO(3)$ — $\mathbb{Z}_n$, $D_n$ and the rotation groups of the tetrahedron, cube and icosahedron; the duality cube–octahedron, icosahedron–dodecahedron; which figures on the sphere realise $\mathbb{Z}_n$, $D_n$ and $\mathbb{Z}_2$.

The order of an element and of a group, the order of a permutation as the lcm of its cycle lengths; generators and cyclic groups; homomorphisms and isomorphisms of groups, the classification of cyclic groups; examples — the sign of a permutation, inversion in an abelian group, the exponential, the determinant, $S_3$ and $D_3$; subgroups, kernel and image, the injectivity criterion; left and right cosets and the partition into them; normal subgroups and conjugation, the index, Lagrange's theorem, subgroups of index 2; the quotient group; the homomorphism theorem with examples; conjugate elements, the centre, a group action on a set.

What geometries there are and what they have in common — Klein's Erlangen programme; the school 3–4–5 triangle and the ellipsoid; the cube and the brick as finite geometries; metric spaces; groups, quantifiers in the axioms; the group of bijections and the order of composition; left and right actions of a group on a set; the metrics $d_1$, $d_2$, $d_\infty$ on $\mathbb{R}^n$; isometries, subgroups; the order of an element; orbits, stabilisers, transitive actions; the list of geometries in the course — the Euclidean plane, the regular triangle and $S_3$, the cube and its 48 symmetries, the Poincaré model of the Lobachevsky plane; fundamental domains; subgeometries — the square in the cube.