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The symmetric group $S_n$ - the group of permutations of an $n$-set - plays a very important role in Group Theory. The paramount importance of this group resides in the following fact: given any finite group $G$, there is a value of n such that $S_n$ possesses a subgroup that is structurally identical with $G$. The above statement is a fact.

Question: What is the minimum value of n such that $S_n$ possesses a subgroup that is structurally identical with the monster group $M$ (respectively other groups $Fi24,B,Co1, Suz$)?

Edit: From the comments. The question is as follows:

Question: Are the minimal degree permutation representations of the sporadic simple groups all known? I am particularly interested in the cases $M, Fi24, B, Co1$ and $Suz$.

Nicky Hekster
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Derak
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    I'm not sure if anybody has studied these particular questions in detail. The groups you're interested in are definitely famous enough that people might have thought about this, but in general the question "what is the minimal $n$ so that $G$ embeds into $S_n$?" is extremely hard, so I wouldn't be surprised to learn there's little to nothing known about your question. – HallaSurvivor Oct 29 '21 at 05:49
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    This 1998 paper states in the abstract that the minimal permutation degree is known for all finite simple groups modulo the classification. So the references therein should answer all your questions. – Arturo Magidin Oct 29 '21 at 06:01
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    @DietrichBurde I think that's a little unreasonable in this case, because they are all instances of the single question "what are the minimal degree permutation representations of the sporadic finite simple groups?". I don't really understand why this question is getting so many close votes. – Derek Holt Oct 29 '21 at 08:05
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    To the voters to close: in what respect is this question unfocused? It could be reworded "Are the minimal degree permutation representations of the sporadic simple groups all known? I am particularly interested in the cases M, Fi24, B, Co1 and Suz." – Derek Holt Oct 29 '21 at 08:29
  • @DerekHolt I have edited the question to make it more attractive. If you (or someone else) think, that I should not edit it this way, please let me know. I will take it back. – Dietrich Burde Oct 29 '21 at 08:42
  • @DietrichBurde Yes that looks better and, as Arturo Magidin said, the minimal degrees of faithful permutation representations of all of the finite simple groups are known. – Derek Holt Oct 29 '21 at 08:44
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    Please use more descriptive titles. – Shaun Oct 29 '21 at 09:44
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    @Shaun Since this question has already been significantly edited, I thought it would be OK to edit the title too! – Derek Holt Oct 29 '21 at 11:05

1 Answers1

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It is more standard to write "isomorphic to" rather than "structurally identical with".

M: 97239461142009186000,

Fi24: 306936,

B: 13571955000,

Co1: 98280,

Suz: 1782.

The ATLAS of Finite Groups is a good source for such information, or its online version

If you looked at their Wikipedia pages you would have found this data for many of these groups.

Incidentally, the Baby Monster B was first proved to exist by Charles Sims, who constructed this representation of degree greater than $13 \times 10^9$ on a computer. There is a now an independent computer-free proof using the construction of M as an automorphism group of the Griess algebra.

Derek Holt
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