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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.

What algebra studies — algebraic structures. Groups (definition, examples, additive notation, uniqueness of the identity and of inverses), rings (variants of the axioms, rings with and without identity), fields and division rings, vector spaces and modules, algebras over a field. The symmetric group $S_n$ — permutations, non-commutativity, two-row notation, cycles and transpositions, decomposition into transpositions, disjoint permutations commute, decomposition into disjoint cycles and its uniqueness, powers, inversions, parity and the sign of a permutation, multiplying by a transposition changes the sign.

Problems for seminar 2 of the course "Algebra 1" (A. A. Avilov, Independent University of Moscow, autumn 2026), 14 September 2026 — group homomorphisms, the homomorphism theorem, conjugate elements, group actions. The group $\operatorname{Hom}(A, B)$ of abelian groups; multiplicativity of the index; the third isomorphism theorem; isomorphic subgroups with non-isomorphic quotients; conjugacy in $S_n$ and cycle type; conjugacy classes of $A_5$; $\operatorname{Aut}(S_n) = S_n$ for $n \ne 6$; the rotation and symmetry groups of the cube; a homomorphism $S_4 \to S_3$; the symmetry group of the icosahedron; the automorphism group of the Petersen graph.

Why homology is needed — the fundamental group as a functor, homotopy groups, bordism and its combinatorial approximation; chain complexes and their homology; abstract simplicial complexes, geometric realisation and triangulations; the complex of simplicial chains, the proof of $d^2 = 0$, the simplicial homology of a segment, of two points and of the boundary of a triangle; reduced homology.