Talk:Sporadic group

How many infinite families of simple groups?

There is a discrepancy between this page and the one dedicated to the classification effort in general. This page refers to the sporadic groups as well as eighteen countably infinite families. The classification page, however, says there are only three such families necessary for the proof. Which figure is correct, or what qualifying information is missing?- —Preceding unsigned comment added by 128.220.30.95 (talk) 23:27, 18 October 2008 (UTC)

The groups of Lie type can be further divided into 16 families. The full list is at list of finite simple groups. Algebraist 13:19, 18 June 2009 (UTC)

Missing an explanatory argument

In representations on the topic of sporadic groups for non-experts (Wikipedia, MathWorld, Ronan) one looks in vain for an explanatory argument (preferably of a heuristic nature) which motivates the fact that there are only finitely many of them. Why, for example, is there no exceptional simple finite group of an order with prime factor larger than 71? --Meerassel (talk) 17:03, 27 January 2010 (UTC)

Table order

Perhaps the table of sporadic group orders should be ordered by order? User4096 (talk) 04:17, 1 May 2010 (UTC)

Since the "Table of the sporadic group orders (w/ Tits group)" is sortable (for several columns), this issue should be solved in principle.

It might, however, be interesting what the initial order is.

Sloane, Wolfram etc. order by order.

Many authors order by "generation" first, then order.

Hiss has Tabelle 2 on page 172, ordered by inventor, invention? –Nomen4Omen (talk) 14:07, 2 March 2023 (UTC)

I was thinking of organizing them by date of discovery, to bring some novelty w/o OR. Else, by type or order would be appropriate too, as you mention, Nomen4Omen. Radlrb (talk) 15:09, 3 March 2023 (UTC)

No simple subquotient in between

The diagram showing the subquotient relations contains the remark "with no simple subquotient in between".
Of course, it is easy to establish that there is no sporadic subquotient in between. But it is not obvious (although almost 100 % probable) that there is no other simple subquotient (e.g. of Lie type) in between. For this a source would be required. Nomen4Omen (talk) 14:14, 2 March 2023 (UTC)

We can check common groups they'd share, this should be easy; there could also be some alternating groups in common that are simple. I don't believe any Lie groups exist inside sporadic groups; that is the very purpose of the Tits group, to provide a link between the two. E₈ is another important link between Lie algebras and the sporadic groups, for example, as are the quasithin groups. Radlrb (talk) 15:13, 3 March 2023 (UTC), updated 04:18, 3 March 2023 (UTC)

Correction, there are Lie algebras as part of different maximal subgroups. There's for example G₂(4) that is a maximal subgroup within Suz. Radlrb (talk) 05:52, 4 March 2023 (UTC), last updated 10:47, 4 March 2023 (UTC).

So I investigated how many Lie groups exist within sporadic groups, in whichever algebraic structure, and found the following result (after a multitude of re-edits):

• Within the first generation of Mathieu groups, there are no groups of Lie type that are maximal subgroups.
• Within the second generation of sporadic groups, only one group has a maximal subgroup of Lie type:

Suz contains: G₂(4)

Another group in this generation contains a simple group of Lie type in its algebraic structure (of a maximal subgroup), but not as a whole:

Co₁ contains : (A₄ × G₂(4)):2

• Within the third generation of sporadic groups, one group has a maximal subgroup that is the only nonstrict group of Lie type:

Fi₂₂ contains: ²F₄(2)'; the Tits group T on its own

Three groups in this generation contain simple groups of Lie type in their algebraic structure, but not as a whole:

Th contains: (3 × G₂(3)):2

B contains: (2² × F₄(2)):2

M contains: 2².²E₆(2²):S₃

• Within the pariahs, one group has a maximal subgroup that is a group of Lie type:

Ly contains: G₂(5)

One group in this class contains a simple group of Lie type in its algebraic structure, but not as a whole:

Ru contains: (2² × Sz(8)):3 or (2² × ²B₂(8)):3

• Worth noting, two groups contain the almost simple double cover of the Tits group as part of their algebraic structure:

Ru (pariah) contains: ²F₄(2) = ²F₄(2)'.2

B contains: S₄ × ²F₄(2)

This is a total of three groups of Lie type (that I know of) that are found inside sporadic groups (when counting T as well), of which none are shared between any of the sporadic groups in whole. Technically, only two groups represent simple Lie groups G₂(4) and G₂(5), with ²F₄(2)' in the grey zone; and some like in M which contains 2².²E₆(2²):S₃, or Ru where one of its maximal subgroups is ²F₄(2), the double cover of the Tits group. Note that G₂ is involved within two different sporadic groups, one of which is a pariah. The only other possible simple subquotients would need to be alternating, some of which are congruent with Classical Chevalley groups A_n(q); I haven't check for these yet (I know A₅ and A₈ appear). I'm going to take a look later, in the meantime maybe also seek that reference we want. o/ 07:52, 4 March 2023 (UTC), 00:17, 5 March 2023 (UTC); last updated Radlrb (talk) 19:30, 23 April 2023 (UTC).

@Radlrb: Thank you so much for your work! As you remark: the non-sporadic Tits group is subquotient of the Rudvalis group. But "in between" means that there would also have to be a sporadic subquotient $X$ of the Tits group, so that we have

${\mathsf {Ru}}\geq {\mathsf {Tits}}\geq X$.           (all groups simple, 2 sporadic)

Is there one? –Nomen4Omen (talk) 16:54, 4 March 2023 (UTC)

My pleasure! @Nomen4Omen: I think it's great that we are discussing this, it's not obvious and maybe hard to find within the literature, so this is good to have. (I moved one more group inside another sub-bullet above, just in case). To answer your question, there is no other sporadic subquotient inside T. Mainly because it is defined within the family of Lie groups, so by definition, the Tits group is organized by what would be generated from a Ree group ²F₄(2²ⁿ⁺¹), none of which have sporadic subquotients (like in general with any of the Lie groups); except for ²F₄(2) which contains the simple index 2 somewhat sporadic subgroup T. There are other simple groups that do lie inside the structure of two maximal subgroups of T: A₆.2² and 5²:4A₄, where A₆ ≃ A₁(9). A₄ is alternating and simple like A₆, where it is isomorphic to the Classical Chevalley A₁(9). A₆.2² and 5²:4A₄ are not, though, simple on their own. So, we can safely say that there are no simple Lie subquotients in between any of the sporadic groups. Ta-da! It is unsurprising, though very pleasant to work out; this surely has been said somewhere and maybe listed in similar fashion, it's just a matter of finding it. Radlrb (talk) 00:17, 5 March 2023 (UTC) (Note: I misread your original question, instead of subquotients in-between sporadic groups, I construed in my mind simply between sporadic groups; see end-comment below of the same day as today... so this result has surely been stated prior. Radlrb (talk) 05:56, 6 March 2023 (UTC)).

$\hookrightarrow$ Okay, so from a search of finite simple alternating groups that are shared between these sporadic groups as maximal subgroups, I have thus found only one example; that is outside the general framework of simple groups that would be shared between sporadic groups from groups that have sporadic subquotients (that are not covers of sporadic groups either), which actually holds some meaning:

The simple alternating group on seven letters, A₇ of group order 2520 is common to three groups as a maximal subgroup:

• M₂₂, a first generation group
• Suz, a second generation group
• O'N, a pariah

M₂₂ and O'N actually hold two copies of these groups which are fused together, while Suz only holds one of these groups. Note too that, of the duplicate alternating groups that are isomorphic with Classical Chevalley groups A_n(q), only A₅ ≃ A₁(4) ≃ A₁(5), A₆ ≃ A₁(9), and A₈ ≃ A₃(2); where A₇ — in between A₅, A₆ and A₈ — does not hold a duplicate isomorphism. Also, an aside, is that 2520 is the first number divisible by the first ten integers 1 through 10. Its Euler totient is 576, which is the square of 24 (incidentally, the sum of its factors 2³ × 3² × 5 × 7 is 24 as well; where 24 comes to play in moonshine and the Leech lattice). This is my addendum, which is notable imho. Other places where factors of 10 come into play inside sporadic groups would be: the irreducible complex representation of M₁₁ is in ten dimensions while its order |M₁₁| = 7920, which is one less than 7919 (the 10³th prime); the order |HS| = 100|M₂₂|, and the irreducible complex dimensional representation of Ly (a pariah) and Th (3rd gen.), respectively, are 2480 and 248. \o/

This example, of course, aside from the subquotients inside a sporadic group that holds other sporadic groups inside. For example, HS holds M₂₂ inside, which in-turn contains A₇ as a maximal subgroup, which means that HS has as a subgroup A₇ deeply inside its structure as well. However, in these three groups that share A₇ (M₂₂, Suz and O'N), none is a subquotient of the other. I'll keep looking a bit more when I come back from work, but I think this is the only alternating group which is shared between groups that are not subquotients within each other. It's actually special! We can wonder what we would do with this; the only maximal subgroup inside M that holds A₇ in any form is the group (A₇ × (A₅ × A₅):2²):2 of order 72,576,000 and a representation of a permutation on 17 points [1]. Radlrb (talk) 03:04, 5 March 2023 (UTC)

$\hookrightarrow$ Update: The simple alternating group on eight letters A₈ of order 20,160 is also common between:

• M₂₃, a first generation group
• Ru, a pariah

This brings the total to only two alternating groups as maximal subgroups (and not simply subroups within a maximal subgroup) common between sporadic groups where one is a not subquotient of another. I checked and double checked, A₇ and A₈ are the only two such groups. It seems noteworthy since through them, they generate a further (more subtle) link between main generation groups and the pariahs. Also, within A₈ (which is the largest member isomorphic to a Classical Chevalley group, A₃(2)) is found A₇, and all smaller alternating groups. Also, A₈ is isomorphic to the second smallest simple non-abelian group projective special linear group PSL(3,2), found inside M₂₂. The largest alternating maximal subgroup that exists within sporadic groups is found inside HN, which is A₁₂. The smallest such group is the (smallest) non-abelian simple group A₅ that exists within J₂, the only non-pariah Janko group.

Therefore, following the above enumeration, there are strictly six finite simple groups that are not sporadic which exist as maximal subgroups inside nine separate sporadic groups; all A or G₂(q). There is also one special case when counting a non-strict group of Lie type, T, which would bring the total to seven such groups inside ten sporadic groups. Listed by order, they are:

• J₂: contains A₅ ≃ A₁(4) order 60
• M₂₂, Suz and O'N: contain A₇ order 2,520
• M₂₃ and Ru: contain A₈ ≃ A₃(2) order 20,160
• Fi₂₂ contains ²F₄(2)', or T, of order 17,971,200
• HN: contains A₁₂ order 239,500,800
• Suz: contains G₂(4) order 251,596,800
• Ly: contains G₂(5) order 5,859,375,000

All of these exist relatively independently of one another and do not generate sporadic subquotients (naturally, by definition). Interestingly, they include members that are for the most part the smallest or smaller members of their respective classes. G₂(3) is notably missing (?).

Worth noting, in line with the showing of finite simple Lie groups that exist within the structure of maximal subgroups (of which four I found, and listed above), there are many alternating groups within the structure of different maximal subgroups of sporadic groups. It would be interesting to quantify these too, and see any common subgroups they'd generate, in particular in the interest of the pariahs, where for example J₃ is most unconnected with the rest of the sporadics; that really is a key here (it might boil down to which cyclic groups are deep down at the root level the most basic connecting elements).

I want to point out that we could have answered the original question on the onset, in the sense that sporadic groups will only have simple sporadic subquotients (for the ones that do); there cannot exist a Lie group or any other simple group which would have inside it a sporadic group, simply by the definition of how they are defined — except for ²F₄(2)′ (also T) which stems from ²F₄(2), with no other simple maximal groups inside either. I misread actually, and sought simple groups shared between any two sporadic group aside from the simple sporadics, and not in-between, i.e. ${\mathsf {G_{a}}}\geq {\mathsf {G_{b}}}\geq {\mathsf {G_{c}}}$. Still, this has been quite a fruitful exercise, so thank you for initiating it @Nomen4Omen. I'll try and find a source for this analysis somewhere, and link it here. I will, for the sake of completion, seek other simple maximal groups that might be in common and list them here, after realizing I had forgotten about isomorphisms which could also be simple (but not finite of Lie type or alternating), of which there shouldn't be that many. This is all in an effort to bring clarity to the structure of the sporadic groups, which readers might find interesting to read here. Radlrb (talk) 05:36, 6 March 2023 (UTC)

"there cannot exist a Lie group or any other simple group which would have inside it a sporadic group, simply by the definition of how they are defined" ...but every group is a subgroup of an alternating group eventually, and also eventually of GL_n(2), or Sp_2n(2), or other "generic" series. --2607:FEA8:F8E1:F00:95F8:E103:5B90:4504 (talk) 03:26, 28 February 2024 (UTC)

Also, if someone wants to edit the diagram to include T then it fits under Fi22 and Ru. (I think there is no need to add a direct link under B. For example, there is no direct link from M22 to M24.) --2607:FEA8:F8E1:F00:95F8:E103:5B90:4504 (talk) 03:50, 28 February 2024 (UTC)

Is the Tits group a sporadic group?

@Radlrb:
Very important sources are:

1. In Eric W. Weisstein, "Sporadic Group", MathWorld, the Tits group is not listed among the 26 sporadic groups.
2. In Eric W. Weisstein, "Tits Group", MathWorld there is a link from this article to the article "Sporadic Group".

To my knowledge, a "link" from one article to a second does NOT have the meaning: The object in the first article IS of the type of the objects in the linked article (the link is only to be understood in a way similar to "see also".) So in my opinion, you have to look for a source for the sporadicity of Tits, a source more specific than this article in mathworld.wolfram.

You added these a while back in May 2018 (1, 2, 3, 4). Having both of these links suggested to me, that it is indeed an ambiguous matter, one which Weisstein is showing here in both ways (as part of, and not as a part of). Radlrb (talk) 22:47, 10 November 2023 (UTC)

Moreover, there has to be given an extremely close look on the attributes defining "sporadic": A finite simple group is called sporadic if

1. it does NOT belong to an infinite systematic family of finite simple groups.

T, as a derivative of a full group of Lie type, is only a root (so to speak, as a zeroth term) in its family of groups -- and importantly so, because this means that there is a slight "sporadic" expression in this family of groups, and sporadic in the sense that it is not generated uniformly from a simple expression that continues to generate groups of the like (i.e. a family) -- so it is not in violation of this, in pure, "strict" terms. The following terms in the set, however are of Lie type, without exception. T is the degenerate case, or special case, or exceptional case, in the infinite family ²F₄(2²ⁿ⁺¹)′ of Lie type for n > 0; sharing some properties of the sporadic groups (more than 1 or 2 maximal subgroups, has multiple semi-presentations, embeds in at least one important group (Fi₂₂), is of relatively large order (larger than four of the Mathieu groups), with its double cover inside other sporadic groups, etc.) Radlrb (talk) 22:06, 10 November 2023 (UTC)

(Although many, many, many infinite systematic families of finite simple groups are of Lie type, this is NOT a defining attribute, because the groups of prime order and the alternating groups also are not of Lie type. But both are infinite systematic families, with the consequence that they do not contain a sporadic element.) The family of commutator groups ²F₄(2²ⁿ⁺¹)′ of Ree groups of type ²F₄ is NOT throughout of Lie type as well. It is, however, an infinite systematic family of finite simple groups. And the Tits group is a member of this infinite systematic family (and thus by definition not sporadic). Nomen4Omen (talk) 15:26, 9 November 2023 (UTC)

I'll link the sources; it's a longstanding understanding that this group specifically has characteristics from both, and many-a-times it has been cited as being in an ambiguous classification, which is why it is classified as such in the literature (as either, both, or neither), from primary and secondary sources. I am busy right now, but I will give you one quick secondary source:

Hartley, Michael I.; Hulpke, Alexander (2010), "Polytopes Derived from Sporadic Simple Groups", Contributions to Discrete Mathematics, 5 (2), Alberta, CA: University of Calgary Department of Mathematics and Statistics: 106, doi:10.11575/cdm.v5i2.61945, ISSN 1715-0868, MR 2791293, S2CID 40845205, Zbl 1320.51021

"The finite simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but there are 26 (or 27 if the Tits group 2F4(2)′ is counted also) which these infinite families do not include."

Also, Mathworld is mainly useful for finding summarized information, and needed sources of interest; it is not a primary source. We should not rely on it as a primary source. Part of the sporadicity of T is found in its simple factorized order, that is in the like of the other small sporadic groups, yet not from the same type of sporadicity, so to speak *while it has a small outer-automorphism (2, as with many sporadic groups), and is also one of seven classes of simple N-groups (which includes M₁₁, as well as A₇, Suzuki groups, and other non-simple groups); this shows that its Lie nature blends with its sporadic nature, from some examples at least* Radlrb (talk) 23:01, 10 November 2023 (UTC). Because most of the literature does acknowledge this generalization of T (really, read into the actual papers), the way it is presented now is PROPER, and ACCURATE. Those references, as I have done for several in our (everyone's) article, I will find, or and if you are inclined to help in this regard, coolio.

And look at this:

Radlrb (talk) 21:00, 10 November 2023 (UTC)

Regarding the bit from Mathworld, I did not want to remove it either because it is not incorrect, though not the best source either (to use as an example in this instance), naturally. Radlrb (talk) 20:53, 11 November 2023 (UTC)

I did a little list of important aspects of T:

• Lie type derivative and root of an infinite series,
  • Only one of its kind (not in a family with other groups lacking a BN-pair, or other of the like),

nor from any other group of Lie type, so exceptional within the family of groups of Lie type, and a group like it does not exist inside families of other simple groups, i.e. cyclic or alternating

• It is one of seven N-groups; which have alternating, sporadic, and other more general types as part

While the alternating group on 7 letters is part of this grouping, the alternating groups on 5, 6, 8 and 9 letters are quasithin groups

• It embeds inside Fi₂₂, and appears as its double cover inside Ru, and B
• Its group order factorization contains a total of four distinct primes (a low number, and minimum amongst the sporadics), otherwise 17 prime numbers, when counting powers separately (a nice association with a count of classes of Lie groups, when including T).

There are other important aspects of T that show its nature blends into both the realm of Lie groups and Sporadic groups; these might be some of the more prominent ones. Radlrb (talk) 02:02, 11 November 2023 (UTC)

---

More precisely,

T turns out to be quite a "meta-expression" of seven (and seventeen) in the classification scheme, so to speak, which gives some reasoning as to why it is of import on the side of sporadic groups:

• It is the seventh largest sporadic group by order, if included as a sporadic group
• It is a middle indexed (order-wise) member in the set of seven Lie groups (when including T as a non-strict group) that are maximal subgroups of sporadic groups (see conclusion from prior discussion, § No simple subquotient in between)
• It contains 13 as its largest prime factor dividing its order: 1 2 3 4 5 6 7 8 9 10 11 12 13 (seven is the middle indexed integer here); with generators (2A, 3, 13),

where also 7 is the first prime number that does not divide its order; any number greater in proportion to 7 (the first being 14, the first integer greater than 13) Radlrb (talk) 01:18, 12 November 2023 (UTC)

• It embeds inside Fi₂₂, which also contains 13 as its largest prime factor dividing its order, with generators (2A, 13, 11)

Fi₂₂ also contains 14 (twice seven) maximal subgroups (two are fused), alongside 69 conjugacy classes (thrice 23)

Looking more carefully, Fi₂₂ is the seventeenth largest sporadic group, sans taking into account T, which embeds inside Fi₂₂, where

• (the seventh prime) 17 is the number of classes of Lie groups if we include T (the seventh composite on the other hand is 14) Radlrb (talk) 02:32, 12 November 2023 (UTC)
• 17 is also the number of primes (inclusive of powers) in the factorization of the order of T,  2¹¹ × 3³ × 5² × 13 = 17,971,200 Radlrb (talk) 03:42, 12 November 2023 (UTC)

• It is one of seven classes of thin groups (when including admissible PSL and PSU groups as general individual categories; ten total otherwise), and one of seven classes of simple N-groups

7 is not arbitrary in the classification, since there are also seven second generation groups, of which J₂ is the smallest (with a uniquely largest p = 7 dividing its order, the smallest maximum prime value dividing the order of any sporadic group), and Suz is the middle indexed by order in this second generation, with a largest prime factor of 13 dividing its order as well, which makes it the only other sporadic group with 13 as the largest prime number dividing its order (after T, and before Fi₂₂) -- (note that, when including T, Suz is also the middle index of all 27 sporadic groups, by order. Radlrb (talk) 03:33, 12 November 2023 (UTC)) In the third generation, the Fischer groups are the series of groups before the higher-realm of two sporadic groups B and F₁; so, T inside Fi₂₂ (that itself fits inside Fi₂₃, and Fi₂₄ in turn) is of great consequence. Worth mentioning, 744 is the number of partitions of 7 into prime parts, where 744 is the constant term in the q-expansion of the j-invariant that represents the infinite graded dimensional representation of F₁. Also, the two smallest-sporadic groups (M₁₁ and M₁₂) do not contain 7 as a prime factor in their order; M₂₂ is the first (non-pariah; J₁ as third largest sporadic group, has an order divisible by 7). The minimal faithful Brauer character of M₂₂ is 21-dimensional, a dimension divisible by 7. Radlrb (talk) 22:45, 11 November 2023 (UTC)

A more basic fact is that it does not fit the order formula. Every finite simple group has order: a prime number, n!/2 for n>4, bunch of polynomials of a prime power blah blah blah, and 27 sporadic integers. --2607:FEA8:F8E1:F00:95F8:E103:5B90:4504 (talk) 03:42, 28 February 2024 (UTC)

Minimal faithful character of the Tits group

The table says that the minimal faithful character of the Tits group is 104, but there are multiple issues with this:

• First and foremost, the character table in the ATLAS of Finite Groups shows 2 26-dimensional representations, exchanged by an outer automorphism, so the minimal faithful character should be 26. The online ATLAS of Finite Group Representations includes these representations too.
• The Tits group is a subgroup of the Rudvalis group Ru and the Fischer group Fi₂₂, which have 28 and 78 dimensional representations, respectively. In the former case, the Tits group actually stabilizes a vector, giving it a 27-dimensional representation.

Before I change this information, why does the table say 104 dimensions in the first place? Blobs2 (talk) 21:45, 30 April 2024 (UTC)