Finite simple groups (non-abelian) can generated by two elements.


Let $G=\langle x,y|x^l=y^m=(xy)^n=1,...\rangle$ be a finite simple group (non-abelian), and $\langle x,y|x^p=y^q=(xy)^r=1,...\rangle$ be another presentation of $G$. *(Here, "..." means possibly more relations).* 


1) If $(1/l)+(1/m)+(1/n)<1$, then does it imply that $(1/p)+(1/q)+(1/r)<1$? 


2) If $(1/l)+(1/m)+(1/n)=1$, then does it imply that $(1/p)+(1/q)+(1/r)=1$? 


3) If $(1/l)+(1/m)+(1/n)>1$, then does it imply that $(1/p)+(1/q)+(1/r)>1$? 


(For a group $G=\langle x,y|x^l=y^m=(xy)^n=1,...\rangle$ to be a group of symmetries of a compact orientable surface, there are some restrictions on the integers $l,m,n$ and genus of surface, due to Riemann-Hurwitz relation. If the integers (l,m,n) satisfy some conditions as above, then it will allow us to consider less number of presentations of finite simple groups to check for possibility of action of the group on the surface).



Presentations of simple groups

> 
> **No**. The simple group of order 60 is a counterexample to (1) and (3).
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If {x,y} is a generating set of G, call its signature 1/|x| + 1/|y| + 1/|xy|. It appears that the signature of most generating sets of non-abelian simple groups are less than 1.


I'll assume that l,m,n are required to be the exact orders (and are not just sneakily enlarged to make it easier to get < 1, since it is already easy to get < 1). Hence you are just asking for some sort of "law of inertia" for the "signature" of two-generator sets.



> 
> However, the simple group of order 60 has the following signatures: 3/5, 11/15, 13/15, 9/10, 31/30.
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It does however appear to be a bit rare. The simple groups I've checked other than A5 all have all signatures less than 1.


I didn't see any simple groups satisfying the hypothesis of (2), so I guess it could be vacuously true.




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Maybe this table of signatures up to order 10000 will help:


* **60 PSL(2,4):** 3/5, 11/15, 13/15, 9/10, 31/30
* **168 PSL(2,7):** 3/7, 15/28, 13/21, 9/14, 61/84, 3/4, 11/14, 17/21, 5/6, 25/28, 11/12, 41/42
* **360 PSL(2,9):** 3/5, 13/20, 7/10, 11/15, 3/4, 47/60, 13/15, 9/10, 11/12, 19/20
* **504 PSL(2,8):** 1/3, 23/63, 25/63, 3/7, 5/9, 37/63, 13/21, 13/18, 95/126, 7/9, 11/14, 17/21, 17/18, 41/42
* **660 PSL(2,11):** 3/11, 23/66, 21/55, 14/33, 151/330, 27/55, 1/2, 17/33, 8/15, 17/30, 13/22, 3/5, 103/165, 2/3, 15/22, 7/10, 11/15, 25/33, 87/110, 5/6, 13/15, 9/10, 61/66
* **1092 PSL(2,13):** 3/13, 27/91, 25/78, 33/91, 211/546, 16/39, 3/7, 19/42, 10/21, 19/39, 1/2, 151/273, 15/26, 13/21, 9/14, 17/26, 2/3, 131/182, 29/39, 11/14, 17/21, 5/6, 71/78, 41/42
* **2448 PSL(2,17):** 3/17, 35/153, 33/136, 43/153, 361/1224, 21/68, 1/3, 25/72, 13/36, 25/68, 3/8, 257/612, 59/136, 23/51, 17/36, 35/72, 1/2, 77/153, 211/408, 5/9, 19/34, 41/72, 7/12, 11/18, 21/34, 5/8, 131/204, 205/306, 93/136, 25/36, 17/24, 13/18, 37/51, 53/72, 3/4, 7/9, 19/24, 55/68, 5/6, 31/36, 7/8, 91/102, 11/12, 17/18, 23/24
* **2520 Alt(7):** 3/7, 19/42, 10/21, 17/35, 1/2, 107/210, 8/15, 15/28, 19/35, 47/84, 17/30, 7/12, 83/140, 3/5, 37/60, 13/21, 9/14, 13/20, 2/3, 71/105, 7/10, 61/84, 11/15, 3/4, 47/60, 11/14, 17/21, 5/6, 59/70, 13/15, 25/28
* **3420 PSL(2,19):** 3/19, 39/190, 37/171, 24/95, 451/1710, 47/171, 3/10, 29/95, 14/45, 29/90, 1/3, 67/190, 311/855, 2/5, 37/90, 19/45, 25/57, 43/95, 277/570, 85/171, 1/2, 23/45, 8/15, 49/90, 5/9, 167/285, 3/5, 23/38, 19/30, 29/45, 62/95, 227/342, 7/10, 32/45, 41/57, 13/18, 11/15, 143/190, 23/30, 7/9, 4/5, 73/90, 13/15, 101/114, 9/10, 14/15, 17/18
* **4080 PSL(2,16):** 3/17, 47/255, 49/255, 1/5, 27/85, 83/255, 1/3, 23/51, 39/85, 7/15, 151/255, 3/5, 21/34, 319/510, 19/30, 37/51, 11/15, 129/170, 23/30, 91/102, 9/10
* **5616 PSL(3,3):** 3/13, 29/104, 25/78, 17/52, 115/312, 3/8, 21/52, 16/39, 5/12, 47/104, 11/24, 19/39, 77/156, 1/2, 167/312, 13/24, 15/26, 7/12, 5/8, 17/26, 103/156, 2/3, 73/104, 17/24, 29/39, 3/4, 19/24, 43/52, 5/6, 71/78, 11/12
* **6048 PSU(3,3):** 1/4, 7/24, 13/42, 1/3, 59/168, 31/84, 3/8, 11/28, 23/56, 5/12, 3/7, 73/168, 19/42, 11/24, 10/21, 1/2, 29/56, 15/28, 13/24, 47/84, 7/12, 101/168, 13/21, 5/8, 9/14, 2/3, 17/24, 61/84, 3/4, 43/56, 11/14, 19/24, 17/21, 5/6
* **6072 PSL(2,23):** 3/23, 47/276, 45/253, 29/138, 661/3036, 57/253, 1/4, 35/138, 17/66, 35/132, 3/11, 27/92, 457/1518, 1/3, 31/92, 15/44, 23/66, 26/69, 389/1012, 5/12, 29/69, 14/33, 19/44, 127/276, 355/759, 1/2, 67/132, 17/33, 25/46, 7/12, 27/46, 13/22, 173/276, 321/506, 2/3, 89/132, 15/22, 49/69, 3/4, 25/33, 73/92, 5/6, 37/44, 121/138, 11/12, 61/66
* **7800 PSL(2,25):** 3/13, 37/156, 19/78, 1/4, 25/78, 17/52, 1/3, 23/65, 281/780, 11/30, 21/52, 16/39, 5/12, 173/390, 9/20, 31/65, 29/60, 19/39, 77/156, 1/2, 137/260, 8/15, 17/30, 15/26, 7/12, 119/195, 37/60, 13/20, 17/26, 103/156, 2/3, 7/10, 29/39, 3/4, 101/130, 47/60, 43/52, 5/6, 71/78, 11/12
* **7920 M11:** 3/11, 27/88, 15/44, 23/66, 3/8, 21/55, 101/264, 183/440, 5/12, 14/33, 19/44, 9/20, 151/330, 11/24, 41/88, 27/55, 59/120, 1/2, 67/132, 17/33, 21/40, 8/15, 119/220, 13/24, 145/264, 17/30, 23/40, 7/12, 13/22, 3/5, 37/60, 103/165, 5/8, 13/20, 79/120, 2/3, 89/132, 15/22, 7/10, 17/24, 63/88, 3/4, 25/33, 47/60, 87/110, 19/24, 33/40, 37/44
* **9828 PSL(2,27):** 3/14, 20/91, 41/182, 3/13, 2/7, 53/182, 27/91, 5/14, 33/91, 3/7, 10/21, 263/546, 19/39, 23/42, 151/273, 13/21, 9/14, 59/91, 17/26, 5/7, 131/182, 31/42, 29/39, 11/14, 17/21, 19/21, 71/78, 41/42


