Monster Group (a little extra bit)

you have the monster here then the second largest one's called the baby monster okay and this can be found inside this can be found inside the monsters as you might expect i think that's got size around 10 to the 33 okay so quite a bit smaller i mean that's the second largest the smallest one so i said the first ones were discovered in the 1860s i think smallest got about size eight thousand just a little bit under eight thousand but but germany the the increase in order they're relatively relatively large okay so i mean this is some sort of like you know tongue-in-cheek pictorial representation of the sort of the classification of finite simple groups so each column in the in the main block here corresponds to one of these infinite families that we talked about which crop up okay so like for example in the very far right hand side here we've got all the groups which have prime size so the size is a prime number so we see c2 here which was like our group s before c3 is like the group r and then we have one for each prime number and then the all the other ones here are the ones which we didn't really explain so these are these are the simple groups which belong to natural infinite families but which do not have prime size okay but then of course because of the existence of the sporadic groups we have this collection down below here which do not fit into any of the which you know do not really belong to this this main table these are the sporadic groups and they're arranged in order of size so at the very end at the very sort of extreme here we have the matthew group m11 so this is one of the ones which is the first one which was discovered in the 1860s that's got order 7920 and then they go up in this direction here we end up here with the monster a lot of the symbols here are sort of the symbols here denote essentially the person who who discovered the group for the first time so m for matthew j fijenko hs for higman sims and so on who discovered the monster not someone called monster no not someone called monster actually no so that's so the baby monster in the monster the only ones i think which don't have that have that property so so the monster the existence of the monster was predicted many years before it actually was was constructed so i think the existence um was conjectured in around 1973 by fisher and grice independently and then it was constructed by grice originally as the symmetry group of this of this object in this enormous dimensional space around about 1980. toronto grass has has he been ripped off by not having his name attached to it well sometimes it's called the fisher grice monster in fact grice actually in his paper called it the friendly giant this was his terminology for but the monster is conway's um terminology i believe and this is this is stuck what's the holy grail of monster group research like what does someone have to find out or publish and say like what's the what's the you know silver bullet paper that you would love to read about the monster group that's a difficult that's a difficult question i mean um i suppose one of the one of the areas where a lot of research has has looked at in recent years is connections with between the monster and other areas of mathematics so there's this there's this idea of these um moonshine phenomena which connects the monster to uh very fundamental objects in number theory on the one hand and mathematical physics on the other and this this is really not fully understood so i think people are sort of interested on the one hand in sort of as a from a group theory point of view just in terms of the properties of this group but also i think the exciting stuff is really these connections with the monster and these other other parts of mathematics

Channel: Numberphile2

Views: 99,215

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Id: MXKiih4JJvQ

Length: 3min 20sec (200 seconds)

Published: Sat May 10 2014