Is this group abelian? – abelian groups detected from the order, from \(G/Z(G)\), and by the structure theorem.
Basics – subgroups, cosets, conjugacy, normal subgroups.
The standard families – cyclic, dihedral, quaternion, symmetric, and alternating groups, and \(p\)-groups.
Quotients, products, and automorphisms – the isomorphism theorems, direct and semidirect products, finitely generated abelian groups.
Groups of small order – the groups of orders \(p\), \(p^2\), and \(pq\), and of each composite order up to \(20\).
Series and solvability – composition series, central and derived series, nilpotent and solvable groups.
Group actions, Sylow theory, and the class equation are in Group actions.