Search results for “ротор”

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.