Semigroups, Monoids and Groups
The ladder from closed operation to semigroup, monoid, group and abelian group with standard examples, uniqueness of identity and inverses, identities and inverses of custom operations, operation-table checks and why a Latin square with an identity can still fail associativity, counts of binary operations, the subgroup test, Lagrange's theorem and its failed converse, element orders, cyclic groups and generators, U(n), S3 and facts that force a group to be abelian.