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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 $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $S_3$, $S_4$.

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.