Groups are studied, starting with basic definitions, like cosets, normal subgroups, quotient groups, normalizers, centralizers, morphisms, leading to the fundamental isomorphisms theorems. On the way, cyclic groups, permutation groups, symmetries, automorphism groups and semidirect products are introduced. Linear groups and their actions on spheres and Grassmannians are examined. Actions, orbits, and conjugacy classes are dealt with in a general context. Several sections are devoted to Sylow theory, and further two on normal series and the theorem of Schreier. Nilpotent and solvable groups are studied with a view towards the simplicity of the alternating group. Free groups are discussed, so are groups given by generators and relations, with e.g. an application to algebraic topology allocated to a less formal section. Transfer homomorphisms and ordered groups are also analyzed.