The classification of finite simple groups was one of the most important problem in group theory. But what makes simple groups so interesting and special?

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    $\begingroup$ See wiipedia, en.wikipedia.org/wiki/Simple_group. Basically, there is a theorem, called the Jordan–Hölder theorem, which says that simple groups are the basic building blocks of group theory. Every group is, in some senses, composed of these groups. So if you know what the finite simple groups are, then you "know" all other finite groups, for a suitable value of "know". $\endgroup$ – user1729 Sep 28 '11 at 13:31
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    $\begingroup$ What makes prime numbers so interesting and special? Since I have class starting in a minute, I'll just post a quote from the Wikipedia page for simple groups: "The finite simple groups are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way prime numbers are the basic building blocks of the integers. This is expressed by the Jordan–Hölder theorem which states that any two composition series of a given group have the same length and the same factors, up to permutation and isomorphism." $\endgroup$ – Alex Sep 28 '11 at 13:31
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    $\begingroup$ Related: math.stackexchange.com/questions/25315/… $\endgroup$ – Bruno Stonek Sep 28 '11 at 13:32
  • $\begingroup$ Simple groups, and their relatives the quasi-simple and almost simple groups, tend to come up a lot. Volume and form preserving matrix groups tend to be quasi-simple, and automorphisms of highly symmetric geometries tend to be almost simple. While simple groups are building blocks in general, the non-abelian ones tend to build very interesting things with only one (non-abelian) block. $\endgroup$ – Jack Schmidt Sep 28 '11 at 14:26

Simple groups are like prime numbers........

One must not let the account of the matter end with a full stop after the last word above. If one could say that all finite groups are products of finite simple groups, then the analogy would be simpler. But one can say that simple groups are as far as you can take the process of taking quotient groups without going to the very smallest quotient group: the group with only one element.

The complication is that just taking Cartesian products of simple groups doesn't give all finite groups, and in fact, I think the understanding the ways in which other finite groups are built out of finite simple groups is quite a substantial problem in itself.

However. Say you have a composite number like $299.$ You can form quotients: $$ 299/13 = 23. $$ $$ 299/23 = 13. $$ But from the prime number $23,$ you can't form any quotients except $23$ and $1.$

Similarly there are no quotient groups of a simple group except itself and the group with only one element.

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