I have finally understood algebra. It is all about understanding groups.(adsbygoogle = window.adsbygoogle || []).push({});

first of all we tackle abelian groups, and finitely generated ones at that., we completely classify them as sums of cyclic groups using their structure as Z modules where Z is the integers.

then we ask about non finitely generated abelian groups. Of course we want to use the same methods that have worked so well before, so we ask if we can view them as modules over some other rings, so that maybe they become finitely generated or cyclic over those rings.

From another point of view, it is natural to ask about the maps of an abelian group as well as the group itself, and the maps from G to G form a ring End(G) = Hom(G,G), such that G is a module over End(G). So we seek to identify a suitable subring R of End(G), such that the structure of G as an R module is simpler, but R is itself not too complicated.

For example, if Q is the rational numbers, then Q^n is not a finitely generated abelian group, but it a finitely generated Q module.

After studying abelian groups as modules over various subrings of End(G), we ask about non abelian groups G. Then we look for rings R such that G is the group of units of R and R acts on some natural modules.

presumably this leads to the analysis of group representatioins of G, and modules over the group ring of G.:!!)

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# Abstract algebra made simple

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