# 25 Summary

We have now come to the end of our discussion of Modern Algebra. We began with binary operations which led us to our first algebraic binary structure, groups. We looked at subgroups, specifically cyclic subgroups. Then, we looked at isomorphisms and how to prove two groups are isomorphic. We also discussed how isomorphims allow us to study more complicated groups by looking at simpler ones because isomorphic groups have the same structural properties. We looked at cosets which were then used to prove the Theorem of Lagrange. We also looked at factor groups, which are special groups whose elements are cosets, and how to find a group that is isomorphic to a particular factor group. We then looked at the FTFGAG, which helps us find a group that is isomorphic to a finite abelian group. Next, we examined a few more algebraic binary structures, known as rings, integral domains, and fields. We concluded with a look at rings of polynomials, reducibility over a field, and irreducibility.

In our next section, we will look at Linear Algebra, a branch of mathematics that has many applications to the real world that is not quite as theoretical as Modern Algebra.