Talk:Monster group

Regarding the change from "a number of" to "26"

Hi, regarding the change from "a number of" to "26", feel free to change, but make sure to explain the change in the edit summary, or here on this talk page. Usually we have so much vandalism from anonymous users, that we revert anything that looks fishy. Also, if you create a username, then it's easier to track your edits and engage in conversations on this. Thanks! Fuzheado 04:42, 4 Feb 2004 (UTC)

I didn't do the original edit, and I don't know that much group theory, but the number 26 also appears in Classification of finite simple groups, so it seems okay to me. Jitse Niesen 11:27, 4 Feb 2004 (UTC)

Hi, that was me, that added the 26. i thought it was more specific that "a number of". i can certainly understand about safeguarding against vandals. next time i will be more informative with my edits -lethe — Preceding unsigned comment added by 144.92.164.199 (talk) 09:28, 5 February 2004 (UTC)

What does "order" mean? The article starts off with an undefined and unlinked word that no layperson is likely to understand. — Preceding unsigned comment added by 211.225.34.183 (talk) 12:34, 1 July 2005 (UTC)

relation to sphere packing

While reading about the Golay code, I was reminded that the Monster had something to do with sphere packing in 24 dimensions; Its the total number of possible different sphere packings or something like that, a finite subgroup of automorphisms in 24 dimensions, or something like that? linas 20:37, 26 May 2006 (UTC)

Hmm, maybe I'm confusing this with the Leech lattice, but I thought that there was more than that.

J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups, Springer-Verlag, New York, Berlin, Heidelberg, 1988.

linas 20:46, 26 May 2006 (UTC)

The novel "Finding Moonshine" contains a nice story about Conway finding the symmetry group of the Golay sphere packing in 24 dimensions by Leech. 85.146.200.37 (talk) 00:44, 4 October 2009 (UTC)

Definition?

What is the definition of "Monster group"? Without this, the uniqueness result is meaningless. Eric119 06:34, 23 August 2006 (UTC)

A historically rooted definition has been supplied. The number 26 is generally accepted by group theorists as the number of sporadic simple groups. Scott Tillinghast, Houston TX (talk) 04:43, 13 April 2008 (UTC)

The diagram

It has been pointed out that M11 is found within the O'Nan group. I do not know how to edit the diagram.

Proposal: Move the O'Nan group along with J1 to the left of the diagram, and add a line from O'N to M11. Move the diagram and Ru a bit to the right to fill the space between Ru and J3. Scott Tillinghast, Houston TX (talk) 01:47, 4 April 2008 (UTC)

Existence and uniqueness

From the section on existence and uniqueness:

Griess first constructed M in 1980 as the automorphism group of the Griess algebra, a 196884-dimensional commutative nonassociative algebra. [...] Griess's construction showed that the Monster existed. John G. Thompson showed that its uniqueness (as a simple group of the given order) would follow from the existence of a 196883-dimensional faithful representation. A proof of the existence of such a representation was announced in 1982 by Simon P. Norton [...]

The page on the Griess algebra mentions that M fixes a non-zero vector in the Griess algebra. So it seems that there exists a faithful 196883-dimensional representation almost by definition? Or was it not yet clear in 1980 that M fixes a non-zero vector in the Griess algebra? I'm a bit confused! Lenny Taelman (talk) 08:56, 24 August 2009 (UTC)

Simple English

I am translating this page into Simple English. Anyone who is proficient in Simple English should feel free to help out. --800km3rk

Sois un pen...dejo la palabra — Preceding unsigned comment added by 2806:1016:6:4D7:2C55:4013:C49:439D (talk) 18:10, 17 June 2018 (UTC)

Please give fuller explanations

Please explain terms a little more. This page presumes that the reader already understands much of its content. I did pursue some of the links of relevant terms, but found only more of the same. (And no, I'm not a dropout. I have taken several math courses, including a few graduate courses.) Thank you very much! 69.247.14.176 (talk) 23:19, 15 December 2014 (UTC)

Hi 69.247.14.176 - Good request. Unfortunately, like many mathematical objects, it is difficult to explain what the monster is in just a few words. I just added What is… The Monster? to the External links section. See if that helps. Also, if you can find a copy of Martin Gardner's Mathematical Games column for June 1980 (also mentioned in the External links), there is a fairly clear description of the monster there.--Foobarnix (talk) 16:52, 5 October 2015 (UTC)

I second the plea of the anonymous reader above. I am also not a "lay reader"; I received a B.Sc. in mathematics from MIT, and my curriculum included a couple of abstract algebra courses. Nevertheless, my earnest attempt to learn what the Fischer-Griess Monster really is from this article did not succeed. One difficulty is the absence of a formal definition; though an earlier comment on the talk page refers to one, it must have been removed by subsequent edits. The text gives several properties of M, and refers vaguely to relationships with other mathematical objects, but never actually defines it in the standard mathematical way. (If "the automorphism group of the Griess Algebra" is intended as a definition, it should be rephrased to make that clearer.)

The link labeled "double cover" leads to an article on double covers of topological groups, which are usually infinite; the definitions there appeal to topological concepts like contiuity, which are not obviously germane to the topic of this article. I think what is meant is a purely algebraic double cover, that is, a group with a homomorphism to B (the baby monster) with Z/2Z as a quotient. But I was unable to find an article about covering groups in a purely algebraic context; I have a suspicion that there once was such an article, but it was the victim of an injudicious merge.

ACW (talk) 14:36, 5 October 2015 (UTC)

factors

The monster's order can also be written 32! · 10! · 4!² · 2² · 7 · 13 · 41 · 47 · 59 · 71. Is that worth mentioning? —Tamfang (talk) 18:28, 5 November 2015 (UTC)

The U3(8) subgroup

Robert A. Wilson has proved that the Monster contains a unique conjugacy class of U3(8) subgroups, see Wilson, Robert A. "The uniqueness of PSU 3 (8) in the Monster." Bulletin of the London Mathematical Society 49.5 (2017): 877-880.

Maximal subgroups

The sentence "The circled symbols denote groups not involved in larger sporadic groups." makes no sense since all symbols in the diagram are circled (unlike in Ronan's book). Benji104 (talk) 05:06, 3 August 2023 (UTC)

Zero-width spaces in 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

The order of the group, “808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000”, is a line-breaking disaster. In a browser of suitable window size, it causes the writing point to jump down about 7cm of space. Ick!

Solution: add a unicode zero-width space after each of the internal commas (except the outermost few). This has been done, but as it is invisible, being flagged here. JDAWiseman (talk) 19:42, 28 October 2023 (UTC)