Talk:Hurwitz's automorphisms theorem

Simple Hurwitz groups

This list of simple groups, examples of Hurwitz groups, is very interesting, but I see no sources. Clearly Hurwitz would not have known of sporadic simple groups except possibly the Mathieu groups.

Is it known what are sufficient conditions for s order 3, t order 2, st order 7 to define a finite group?

Are there sporadic Hurwitz groups, or just sporadic simple groups that happen to be Hurwitz groups? Scott Tillinghast, Houston TX (talk) 22:29, 13 March 2008 (UTC)

• I am not sure I understand your question exactly, but perhaps the following comments will clarify the picture: 1. Hurwitz himself never studied the so-called Hurwitz groups. This terminology was introduced much later. 2. There exists a classification of finite simple groups, including a finite number of infinite families, as well as a few additional examples (finitely many of them). The latter are called sporadic or exceptional finite simple groups. 3. The question which of the finite simple groups happen to arise as Hurwitz groups, i.e. finite quotients of the (2,3,7) triangle group, has stimulated a lot of research but there is as yet no complete answer as far as I know. Concerning some of the sporadic finite simple groups, the question has been answered (see the article itself). 4. As far as I know, there is no independent notion of a "sporadic Hurwitz group; at any rate it is clear that Hurwitz himself did not introduce any such notion. Katzmik (talk) 14:10, 18 March 2008 (UTC)

Commutator order

I was wondering about these groups before I knew they were called Hurwitz groups. The 3 conditions s order 3, t order 2, st order 7 of course define an infinite group. I have wondered whether a commutator [s,t]=stsst of finite order is sufficient to define a finite group.

• [s,t] order 2: not compatible with st order 7
• [s,t] order 4: simple, order 168
• [s,t] order 9: simple, order 504
• [s,t] order 11: first Janko, order 175560
• [s,t] order 12: Hall-Janko, order 604800
• [s,t] order 13: PSL(2,27), order 13*27*28
• [s,t] order 15: G2(3), order in the billions
• [s,t] order 19: first Janko, order 175560

I have found these on Robert A. Wilson's website. It does not consistently list complete sufficient conditions for defining the groups.

Yes, I know about the sporadic simple groups, of which only the Mathieu groups were known in Hurwitz's lifetime. Scott Tillinghast, Houston TX (talk) 00:59, 22 March 2008 (UTC)

Letting H(r) = < a,b : a^2=b^3=(ab)^7=[a,b]^r=1 > be the finitely presented group whose only extra relation beyond the (2,3,7) triangle group relations is an order requirement on the commutator, one has the following results:

r	order(H(r))	H(r)
1	1	trivial group (so <a,b:a^2=b^3=(ab)^7=1> is perfect)
2	1	trivial group
3	1	trivial group
4	168	PSL(2,7)=PSL(3,2)
5	1	trivial group
6	1092	PSL(2,13)
7	1092	PSL(2,13)
8	10752	PSL(3,2) N 2^3 x N 2^3', nonsplit extension of PSL(3,2) by the direct product of its natural module and its dual
9	∞	has a quotient isomorphic to the infinite perfect group PSL(2,8) N Z^7 N 2^1 has a quotient isomorphic to the nonsplit extension of PSL(2,8) by a module extension of its natural module by the trivial module, aka, PSL(2,8) N 2^6 E 2^1
10	∞	has a quotient isomorphic to the infinite perfect group (PSL(2,41) x J1 x J1 x J2 x J2) N Z^42
11	≥ 175560^2 · 39732 · 43^(11+14+14)	has a quotient isomorphic to (PSL(2,43) x J1 x J1) N 43^11 N 43^14 N 43^14
12	≥ 604800 · 168 · 3^(6+14+49+189) · 1092 · 2^(28 + 314)	has a quotient isomorphic to (PSL(3,2) x PSL(2,13) x J2) N ( (3^6 N 3^14 N 3^49 N 3^189) x (2^28 N 2^314) )

In other words, simply specifying the commutator order is not sufficient to determine the finite simple groups that occur as quotient groups. Sometimes the resulting group is infinite, and sometimes it has more than one simple quotient. It seems likely to me that very few of the H(r) are simple. I am not very confident about whether most H(r) are finite or infinite. I think one should check exactly which PSL(2,q) are Hurwitz and for which "r" they occur. JackSchmidt (talk) 18:44, 22 March 2008 (UTC)

Sources?

What is the source for the statement that the 12 sporadic groups are Hurwitz groups? Scott Tillinghast, Houston TX (talk) 02:44, 22 March 2008 (UTC)

I've found a few papers whose main purpose is to show a specific sporadic group is a Hurwitz group. If you have access to MathSciNet, then you can probably find sources fairly quickly. Any large university should have access even from the library card catalog computers. I don't know of a reference for all of them, but here is an interesting excerpt from a recent math review :

“ Many families of groups have been shown to have this property, including the alternating groups A_n for all n>167, the groups PSL(2,q) for certain q, the groups SL(n,q) for all n >= 287 and every prime-power q, many of the exceptional simple groups of Lie type, and 12 of the 26 sporadic finite simple groups. On the other hand, for example, it is known that very few of the groups PSL(3,q) and PSL(4,q) are Hurwitz. ”

Here is the second most recent paper from a simple such search:

• Wilson, Robert A. (2001), "The Monster is a Hurwitz group.", J. Group Theory, 4 (4): 367–374, doi:10.1515/jgth.2001.027, MR 1859175

I have to run, but if you want some more specific citations, they are not too hard to get. I'm also planning on doing a few checks on your interesting commutator question. JackSchmidt (talk) 16:06, 22 March 2008 (UTC)

Degree of transitive rep of Hurwitz group

What is the source for the statement about alternating groups? I have experimented and found that the smallest such Hurwitz group may be A14. The relations are not even possible for transitive groups of degrees 10 through 13. There is essentially one for degrees 7 and 8 (order 168) and degree 9 (order 504). Scott Tillinghast, Houston TX (talk) 02:44, 22 March 2008 (UTC)

The group of the first Hurwitz triplet has order 13*84, which is right for PSL(2,13). That group has a permutation representation of degree 14. Scott Tillinghast, Houston TX (talk) 05:07, 22 March 2008 (UTC)

Here is the reference for the large degree alternating groups:

• Conder, Marston D. E. (1980), "Generators for alternating and symmetric groups", Journal of the London Mathematical Society, 22 (1): 75–86, doi:10.1112/jlms/s2-22.1.75, ISSN 0024-6107, MR 0579811

This paper also claims that the degree of a transitive permutation representation of the (2,3,7) triangle group must be of the form 84*(p-1)+21*e+28*f+36*g for non-negative integers p,e,f,g. This is obviously false, but perhaps you might find the paper interesting and might fix the claim.

I confirm computationally that there are no transitive reps of degree 2,3,4,5,6,10,11,12,13, only one of degrees 7,8,9, and for n=14 there are 3 transitive Hurwitz groups of degree 14, and about six transitive permutation representations (so some groups have more than one conjugacy class of representation). The Hurwitz groups that are also transitive groups of degree 14 are PSL(2,7), PSL(2,7) N 2^3, and PSL(2,13). JackSchmidt (talk) 19:22, 22 March 2008 (UTC)

Degree 15 just finished as I hit save. The only Hurwitz group that is also a transitive group of degree 15 is the alternating group of degree 15. The alternating group of degree 14 is not a Hurwitz group. Degree 16 is a little too big to brute force, but let me know if you are interested in such low degree calculations as several other small degrees are still easy. JackSchmidt (talk) 19:32, 22 March 2008 (UTC)

Thank you. So A15 is the smallest alternating group that is a Hurwitz group.

Now we can consider what to add to the article. A footnote referring to Wilson's paper on the Monster sounds good to me. Scott Tillinghast, Houston TX (talk) 00:28, 24 March 2008 (UTC)

Symmetries of elliptic curves

"in fact, the conformal automorphism group is a connected complex Lie group of dimension three for a sphere and of dimension one for a torus"

Is "connected" true? Can't an elliptic curve have extra symmetries, like $\mathbb {C} /\Gamma$, where Γ is a rectangular or hexagonal lattice? --129.132.146.194 (talk) 09:06, 18 August 2010 (UTC)

You are right! Since the importance of the parenthetical comment is in the positive dimensionality of the automorphism group for genus zero or one, I have simply removed the connectedness claim. Arcfrk (talk) 06:16, 19 August 2010 (UTC)

"Remarkable fact" not so remarkable?

The article had said:

"Thus we are asking for integers which make the expression

1 − 1/p − 1/q − 1/r

strictly positive and as small as possible. A remarkable fact is that this minimal value is 1/42..."

Referring to this fact as 'remarkable' may signal it as a nontrivial result, whereas it is (in the context of such a proof sketch) quite unremarkable.

For completeness: Bearing in mind the result must be strictly positive, the greatest unit fraction that can be subtracted off of 1 is 1/2; the remainder, 1/2, can then have at most 1/3 subtracted off; this yields 1 - 1/2 - 1/3 = 1/6, from which at most 1/7 can be subtracted.

This gives 1 - 1/2 - 1/3 - 1/7 = 1/42 as claimed. — Preceding unsigned comment added by 209.6.139.62 (talk) 04:14, 23 June 2015 (UTC)

That is close to a proof, but something needs to be said about why the greedy algorithm results in the absolute minimum. 2601:200:C000:1A0:813D:6802:379E:94B1 (talk) 22:10, 15 October 2022 (UTC)

Title wrong?

Shouldn't this page be called "Hurwitz's automorphism theorem"? I find it hard to believe people talk about "Hurwitz's automorphisms theorem", even though it concerns more than one automorphism. What do most mathematicians actually call this theorem? John Baez (talk) 20:54, 5 December 2015 (UTC)

I can assure you that what mathematicians call it is "Hurwitz's theorem". Or if the context is not clear, "Hurwitz's 84(g-1) theorem". 2601:200:C000:1A0:813D:6802:379E:94B1 (talk) 22:13, 15 October 2022 (UTC)

"Hurwitz" surfaces?

I did not know the term "Hurwitz surfaces" for those Riemann surfaces for which Hurwitz's inequality is actually an equality.

But of course some individual examples have their own names, like the unique example of least genus: the Klein quartic of genus 3. Also, one entire infinite family of examples has been named "Macbeath surfaces", after their discoverer.

So, I wonder if someone could include in the article the origin of the term "Hurwitz surfaces", such as who coined the term and when. 22:03, 15 October 2022 (UTC)

Cuboctahedron immersion

I am trying to see how the illustration of the cuboctahedron immersion could be the Klein quartic tiled with 56 triangles meeting (as they must) 7 per vertex.

It would be most helpful if the creator of that graphic — or someone else — could append the caption with an explanation.

OR — could this be an error, a misidentification? I suspect so. 2601:200:C000:1A0:813D:6802:379E:94B1 (talk) 04:09, 16 October 2022 (UTC)