Lectures 18&19
Permutations: as a composition of disjoint cycles; as a composition of transpositions. Definition of odd & even permutations.
Congruences and left cosets: Proved that left cosets have the same number of elements and that they partition the group. Lagrange's theorem and the fact that the order of an element divides the order of the group.
Definition of the order of an element in terms of the cyclic subgroup the element generates. I also gave some examples using G=S_3 and H= cyclic subroup of order 3 (finding the left cosets, the order of some elements etc). Decomposing a partition into transpositions to determine parity.