Group theory 14: Sylow theorems

This lecture is part of an online mathematics course on group theory, focusing on the Sylow theorems. The speaker begins by motivating the theorems with the question of whether a group has a subgroup of every order dividing its order, noting that this is not always true (e.g., rotations of a tetrahedron have order 12 but no subgroup of order 6). He then states the Sylow theorems: existence of a Sylow p-subgroup (of order the largest power of p dividing the group order), the number of such subgroups is congruent to 1 mod p and divides the group order, all Sylow p-subgroups are conjugate, and any p-subgroup is contained in a Sylow p-subgroup. The proof of existence uses induction on the group order, with a case analysis based on the index of proper subgroups. The proof of the number and conjugacy uses group actions on the set of Sylow p-subgroups, showing that orbits have sizes divisible by p except for the fixed point. Finally, the theorem that any p-subgroup is contained in a Sylow p-subgroup is proved by contradiction using maximality. The lecture concludes with an example of the dihedral group D4 (corrected to D8) to illustrate that subgroups of order p^2 need not be conjugate if p^2 is not the maximal power. The next lecture will use Sylow theorems to classify groups of order 12.