Theorem:

Every abelian group is the direct product of its Sylow subgroups.

Perhaps it is not the theorem in itself I like so much but what this theorem illustrates about the nature of mathematics. Most laymen think of mathematics as the scribbles of physicists they see in science documentaries, i.e. partial differential equations, stuff they call 'formulas'. So in that sense the theorem above may not even be recognized as mathematics, let alone beautiful mathematics.

Mathematics starts with a very precise, razor blade sharp, use of the tool that differentiates us humans from the rest of nature: language. Einstein once said “If you can't explain it to a six year old, you don't understand it yourself.” (*). He must have meant the "root of your knowledge tree", I suppose. Because the beauty of the theorem lies in what it represents: a large graph of concepts with - ( abelian ) group, direct product and ( Sylow ) subgroup - in the center. To anyone 'owning' these concepts the particular relation between an abelian group and its Sylow subgroups can be described in one sentence with no room whatsoever for misinterpretation. The construction of all that knowledge is the collective work of thousands and thousands of mathematicians before us.

P.S.

(*) The simplest way to explain a group is ( as far as I know ) "A collection of movements with no visible effects ( = symmetries )".

Both Abel and Sylow were Norwegians. So was Lie, another giant, a special branch in group theory is named after him: Lie Group Theory. It is amazing that a small country like Norway ( measured in population ) can have such an impact.

I find the analysis of a group very interesting.

ReplyDeleteThis is a great theorem!

ReplyDeleteOnly great minds could see them!

Gabions

Really beautiful, many people don't understand how much such a short sentence can say. Group theory is really something I enjoyed studying.

ReplyDeletePls give a detailed information about theory..

ReplyDeleteThanks for the Information

Regards

Education Portal

The posts are seems ENDEVOUR.....

ReplyDeleteThanks for the Information

Regards

Education Portal

Remember there is a post soviet effect. Which put a lot of emphasize in Science and Mathematics. And Norway, is not far from 'Ionian' Greece/Turkey or least near to that area. I heard a lot about 'Hungarian Mathematics'. I don't know what does it mean. But I here, that aspiring mathematicians who want to specialize in 'combinatorics' practice 'Combinatorial Problems and Exercises' for years. So people in those cultures are destined to be good at math :)

ReplyDeleteThis comment has been removed by a blog administrator.

ReplyDelete