Elementary Divisors and Invariant Factors (Algebra 1: Lecture 19 Video 1)

Lecture 19: We started this lecture by talking more about how to go back and forth between the Invariant Factor Decomposition and the Elementary Divisor Decomposition of a finite abelian group G.  We then discussed 'Recognizing Direct Products'.  We recalled some results about the product set HK and in particular, we needed a result about the number of ways that each element of HK can be written as a product of something in H times something in K.  We then stated and proved the 'Recognition Theorem for Direct Products'.

Reading: We continued our discussion of invariant factors and elementary divisors from the last lecture, following Section 5.2 of Dummit and Foote. We have now covered everything in that section.  We skipped over Section 5.3 for now and instead went through some of the material of Section 5.4.  We did not discuss Proposition 7 about commutators, but you may want to read over it.  We discussed Proposition 8 and how it was related to results from Section 3.2 that we proved much earlier.  We then stated as proved Theorem 9.  We will begin the next lecture by discussing the three examples at the end of this section.