# ALGEBRAIC COMBINATORICS

On prime order automorphisms of generalized quadrangles
Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 143-160.

In this paper, we study prime order automorphisms of generalized quadrangles. We show that, if $\mathrm{đŹ}$ is a thick generalized quadrangle of order $\left(s,t\right)$, where $s>t$ and $s+1$ is prime, and $\mathrm{đŹ}$ has an automorphism of order $s+1$, then

 $sââ\frac{{t}^{2}}{s+1}â\left(\frac{s+1}{t}\right)ââ€t\left(s+t\right),$

with a similar inequality holding in the dual case when $t>s$, $t+1$ is prime, and $\mathrm{đŹ}$ is a thick generalized quadrangle of order $\left(s,t\right)$ with an automorphism of order $t+1$.

In particular, if $s+1$ is prime and if there exists a natural number $n$ such that

 $\frac{{t}^{2}}{n+1}+tâ€s+1<\frac{{t}^{2}}{n},$

then a thick generalized quadrangle $\mathrm{đŹ}$ cannot have an automorphism of order $s+1$, and hence the automorphism group of $\mathrm{đŹ}$ cannot be transitive on points. These results apply to numerous potential orders for which it is still unknown whether or not generalized quadrangles exist, showing that any examples would necessarily be somewhat asymmetric. Finally, we are able to use the theory we have built up about prime order automorphisms of generalized quadrangles to show that the automorphism group of a potential generalized quadrangle of order $\left(4,12\right)$ must necessarily be intransitive on both points and lines.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.89
@article{ALCO_2020__3_1_143_0,
author = {Afton, Santana F. and Swartz, Eric},
title = {On prime order automorphisms of generalized quadrangles},
journal = {Algebraic Combinatorics},
pages = {143--160},
publisher = {MathOA foundation},
volume = {3},
number = {1},
year = {2020},
doi = {10.5802/alco.89},
zbl = {07169927},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.89/}
}
Afton, Santana F.; Swartz, Eric. On prime order automorphisms of generalized quadrangles. Algebraic Combinatorics, Volume 3 (2020) no. 1, pp. 143-160. doi : 10.5802/alco.89. https://alco.centre-mersenne.org/articles/10.5802/alco.89/

[1] Adm, Mohammad; Bergen, Ryan; Ihringer, Ferdinand; Jaques, Sam; Meagher, Karen; Purdy, Alison; Yang, Boting Ovoids of generalized quadrangles of order $\left(q,{q}^{2}-q\right)$ and Delsarte cocliques in related strongly regular graphs, J. Combin. Des., Volume 26 (2018) no. 5, pp. 249-263 | Article | MR 3782235 | Zbl 1400.05284

[2] Bamberg, John; Giudici, Michael; Morris, Joy; Royle, Gordon F.; Spiga, Pablo Generalised quadrangles with a group of automorphisms acting primitively on points and lines, J. Combin. Theory Ser. A, Volume 119 (2012) no. 7, pp. 1479-1499 | Article | MR 2925938 | Zbl 1245.05014

[3] Bamberg, John; Glasby, Stephen Peter; Popiel, Tomasz; Praeger, Cheryl E. Generalized quadrangles and transitive pseudo-hyperovals, J. Combin. Des., Volume 24 (2016) no. 4, pp. 151-164 | Article | MR 3487142 | Zbl 1338.05031

[4] Bamberg, John; Li, Cai Heng; Swartz, Eric A classification of finite antiflag-transitive generalized quadrangles, Trans. Amer. Math. Soc., Volume 370 (2018) no. 3, pp. 1551-1601 | Article | MR 3739185 | Zbl 1381.51001

[5] Bamberg, John; Li, Cai Heng; Swartz, Eric A classification of finite locally 2-transitive generalized quadrangles (2019) (https://arxiv.org/abs/1903.07442)

[6] Bamberg, John; Popiel, Tomasz; Praeger, Cheryl E. Point-primitive, line-transitive generalised quadrangles of holomorph type, J. Group Theory, Volume 20 (2017) no. 2, pp. 269-287 | Article | MR 3619129 | Zbl 1428.20004

[7] Bamberg, John; Popiel, Tomasz; Praeger, Cheryl E. Simple groups, product actions, and generalized quadrangles, Nagoya Math. J., Volume 234 (2019), pp. 87-126 | Article | MR 3938840 | Zbl 07049014

[8] Benson, Clark T. On the structure of generalized quadrangles, J. Algebra, Volume 15 (1970), pp. 443-454 | Article | MR 0289332 | Zbl 0212.52304

[9] Bouniakowsky, Victor Nouveaux thĂ©orĂšmes relatifs Ă  la distinction des nombres premiers et Ă  la composition des entiers en facteurs, Sc. Math. Phys., Volume 6 (1857), pp. 305-329

[10] Cameron, Peter J. Permutation groups, London Mathematical Society Student Texts, 45, Cambridge University Press, Cambridge, 1999, x+220 pages | Article | MR 1721031 | Zbl 0922.20003

[11] De Winter, Stefaan; Kamischke, Ellen; Wang, Zeying Automorphisms of strongly regular graphs with applications to partial difference sets, Des. Codes Cryptogr., Volume 79 (2016) no. 3, pp. 471-485 | Article | MR 3489753 | Zbl 1336.05086

[12] De Winter, Stefaan; Thas, Koen Generalized quadrangles with an abelian Singer group, Des. Codes Cryptogr., Volume 39 (2006) no. 1, pp. 81-87 | Article | MR 2201385 | Zbl 1172.51301

[13] De Winter, Stefaan; Thas, Koen The automorphism group of Payne derived generalized quadrangles, Adv. Math., Volume 214 (2007) no. 1, pp. 146-156 | Article | MR 2348027 | Zbl 1129.51005

[14] De Winter, Stefaan; Thas, Koen Generalized quadrangles admitting a sharply transitive Heisenberg group, Des. Codes Cryptogr., Volume 47 (2008) no. 1-3, pp. 237-242 | Article | MR 2375470 | Zbl 1185.51007

[15] Gavrilyuk, Alexander L.; Makhnev, Aleksander A. On automorphisms of a distance-regular graph with intersection array $\left\{56,45,1;1,9,56\right\}$, Dokl. Akad. Nauk, Volume 432 (2010) no. 5, pp. 583-587 | Article | MR 2766516

[16] Ghinelli, Dina Regular groups on generalized quadrangles and nonabelian difference sets with multiplier $-1$, Geom. Dedicata, Volume 41 (1992) no. 2, pp. 165-174 | Article | MR 1153980 | Zbl 0746.51011

[17] Gill, Nick Transitive projective planes, Adv. Geom., Volume 7 (2007) no. 4, pp. 475-528 | Article | MR 2360899 | Zbl 1139.51014

[18] Gill, Nick Transitive projective planes and insoluble groups, Trans. Amer. Math. Soc., Volume 368 (2016) no. 5, pp. 3017-3057 | Article | MR 3451868 | Zbl 1345.20005

[19] Huppert, Bertram; Lempken, Wolfgang Simple groups of order divisible by at most four primes, Proc. F. Scorina Gomel State Univ., Volume 16 (2000) no. 3, pp. 64-75 | Zbl 1159.20303

[20] Isaacs, I. Martin Finite group theory, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, RI, 2008, xii+350 pages | Article | MR 2426855 | Zbl 1169.20001

[21] MaÄaj, Martin; Ć irĂĄĆ, Jozef Search for properties of the missing Moore graph, Linear Algebra Appl., Volume 432 (2010) no. 9, pp. 2381-2398 | Article | MR 2599868 | Zbl 1217.05149

[22] Makhnev, Alexander A. Jr.; Makhnev, Alexander A. Ovoids and bipartite subgraphs in generalized quadrangles, Mat. Zametki, Volume 73 (2003) no. 6, pp. 878-885 | Article | MR 2010657 | Zbl 1056.51004

[23] OâKeefe, Christine M.; Penttila, Tim Automorphism groups of generalized quadrangles via an unusual action of $\text{P}\mathrm{Î}\text{L}\left(2,{2}^{h}\right)$, European J. Combin., Volume 23 (2002) no. 2, pp. 213-232 | Article | MR 1881553 | Zbl 1028.51006

[24] Payne, Stanley E. An inequality for generalized quadrangles, Proc. Amer. Math. Soc., Volume 71 (1978) no. 1, pp. 147-152 | Article | MR 0476547 | Zbl 0357.05027

[25] Payne, Stanley E. The fundamental theorem of $q$-clan geometry, Des. Codes Cryptogr., Volume 8 (1996) no. 1-2, pp. 181-202 (Special issue dedicated to Hanfried Lenz) | Article | MR 1393984 | Zbl 0874.51001

[26] Payne, Stanley E.; Thas, Joseph A. Finite generalized quadrangles, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), ZĂŒrich, 2009, xii+287 pages | Article | MR 2508121 | Zbl 1247.05047

[27] Praeger, Cheryl E. An OâNanâScott theorem for finite quasiprimitive permutation groups and an application to $2$-arc transitive graphs, J. London Math. Soc. (2), Volume 47 (1993) no. 2, pp. 227-239 | Article | MR 1207945 | Zbl 0738.05046

[28] Swartz, Eric On generalized quadrangles with a point regular group of automorphisms, European J. Combin., Volume 79 (2019), pp. 60-74 | Article | MR 3899084 | Zbl 1419.51004

[29] Tits, Jacques Sur la trialitĂ© et certains groupes qui sâen dĂ©duisent, Inst. Hautes Etudes Sci. Publ. Math., Volume 2 (1959), pp. 14-60 | Article | Zbl 0088.37204

[30] Yoshiara, Satoshi A generalized quadrangle with an automorphism group acting regularly on the points, European J. Combin., Volume 28 (2007) no. 2, pp. 653-664 | Article | MR 2287459 | Zbl 1111.51006