Where do these enormous sporadic simple groups come from?

Question

I mean, there are simple groups of big order such as 808017424794512875886459904961710757005754368000000000

I think it's order is something similar to a factorial for all those 0s... but I'd like to know how were these built...

I assume you're referring to the sporadic groups. There are finite simple groups of arbitrarily large order. — Matt Samuel, Mar 08 '15 at 19:50
Related: What is the simplest way to fathom the Monster Group?, Why is the Monster group the largest sporadic finite simple group?. — MJD, Mar 08 '15 at 19:51
I think this q/a on MathOverflow would be helpful: http://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable . — zibadawa timmy, Mar 08 '15 at 19:57
recommend http://www.maa.org/publications/maa-reviews/from-error-correcting-codes-through-sphere-packings-to-simple-groups — Will Jagy, Mar 08 '15 at 20:40

score 5 · Accepted Answer · edited Jan 18 '19 at 16:57

(long for a comment..)

If you refer to the monster group, then I would suggest you this http://youtu.be/jsSeoGpiWsw - if John Conway says that he doesn't know why the monster group exists, I doubt that anyone can give a reasonable answer. In general, Matt Samuel's comment above is right, it is enough to consider a finite simple group of classical Lie type (i.e. certain matrices of size $n$ over a certain field) when $n$ goes to infinity the order of the group goes to infinity.

If you want a reason for why the study of finite simple groups is become important (from a theoretical point of view) then you should have a look at the Jordan-Hölder theorem, which states that every finite group is built up by a finite number of finite simple groups. Hence the finite simple groups are the elements of the 'periodic table of group theory'.

Where do these enormous sporadic simple groups come from?

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