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