Talk:Hurwitz's automorphisms theorem

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[edit] 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)

[edit] 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)

[edit] 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:
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)

[edit] 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:
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)