Wednesday Nov 15

Lagrange's thm. A₄ violates the converse statement.
p^k || m.
Def. Sylow subgroup, Syl_p(G).
Statement of Sylow's theorems.
Lemma. p doesn't divide (m choose p^k).
Lemma. If H,N finite subgroups of G, then |HN| = formula, and if H normalizes N, then HN is a subgroup.
Thm. Sylow subgroups exist. Pf. By acting on p^k-element subsets by left mult.
Thm. When S acts on Syl_p by conjugation, the only fixed point is itself.