Solvable Groups (Algebra 1: Lecture 7 Video 5)

Lecture 7: In this lecture we discussed the Third Isomorphism Theorem (the 'invert and cancel' one) and then the Fourth/Lattice Isomorphism Theorem.  We gave some examples and proved some similar results, for example, that the 'Index is Multiplicative'.  We then discussed the idea that if N is a normal subgroup of G and you know a lot about N and about G/N, sometimes you can use this information to prove things about G.  As an example we proved Cauchy's Theorem for finite abelian groups.  We then defined Composition Series and stated (but did not prove) the Jordan-Hölder theorem.  We defined what it means for a group to be solvable, and proved that if N is a normal subgroup of G with both N and G/N solvable, then G is solvable as well.Reading: In this lecture we started by following the end of Section 3.3 pretty closely.  We then started Section 3.4.  We proved Proposition 21 and stated Theorem 22 before briefly discussing The Hölder Program (see page 103).  We gave the definition of Solvable from page 105.  The proposition we proved in the last video is covered in the paragraph after the proof of Hall's theorem on that page.