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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.