Proof of the Fundamental Theorem of Finite Abelian Groups (Algebra 1: Lecture 20 Video 3)

Lecture 20: We started this lecture by discussing some applications of the Recognition Theorem for Direct Products. The first application was to certain dihedral groups and the other two were to symmetric groups. We then showed how to use the Recognition Theorem for Direct Products to prove the Primary Decomposition Theorem that we discussed two lectures ago. (We did not prove the uniqueness part of the theorem.) The big step was to prove a lemma about a finite abelian group G of order p^n decomposing as G \cong (g) x H where g in G is an element of maximal order.