ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.04
https://doi.org/10.26493/1855-3974.2507.a1d
(Also available at http://amc-journal.eu)
A parametrisation for symmetric designs
admitting a flag-transitive, point-primitive
automorphism group with a product action*
Eugenia O’Reilly-Regueiro †
Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de
la Investigación Científica, Circuito Exterior, Ciudad Universitaria, Coyoacán, 04510,
Mexico City, Mexico
José Emanuel Rodríguez-Fitta
Facultad de Estudios Superiores Acatlán, Universidad Nacional Autónoma de México
(UNAM), Av. Alcanfores y San Juan Totoltepec s/n, Santa Cruz Acatlán, Naucalpan,
53150, Edo. de México, Mexico
Received 17 December 2020, accepted 2 May 2022, published online 11 November 2022
Abstract
We study (v, k, λ)-symmetric designs having a flag-transitive, point-primitive automor-
phism group, with v = m2 and (k, λ) = t > 1, and prove that if D is such a design with
m even admitting a flag-transitive, point-primitive automorphism group G, then either:
(1) D is a design with parameters
(
(2t+ s− 1)2, 2t
2−(2−s)t
s ,
t2−t
s2
)
with s ≥ 1 odd, or
(2) G does not have a non-trivial product action.
We observe that the parameters in (1), when s = 1, correspond to Menon designs.
We also prove that if D is a (v, k, λ)-symmetric design with a flag-transitive, point-
primitive automorphism group of product action type with v = ml and l ≥ 2 then the
complement of D does not admit a flag-transitive automorphism group.
Keywords: Symmetric-designs, flag-transitivity, primitive groups, automorphism groups of designs.
Math. Subj. Class. (2020): 05B05, 51E05, 20B15, 20B25
*The authors would like to express their gratitude to the referee who made very helpful comments and sugges-
tions that improved our paper.
†Corresponding author.
E-mail addresses: eugenia@im.unam.mx (Eugenia O’Reilly-Regueiro), 887191@pcpuma.acatlan.unam.mx
(José Emanuel Rodríguez-Fitta)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.04
https://doi.org/10.26493/1855-3974.2507.a1d
(Dostopno tudi na http://amc-journal.eu)
Parametrizacija simetričnih dizajnov, ki
dopuščajo praporno tranzitivno in točkovno
primitivno grupo avtomorfizmov s produktnim
delovanjem*
Eugenia O’Reilly-Regueiro †
Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de
la Investigación Científica, Circuito Exterior, Ciudad Universitaria, Coyoacán, 04510,
Mexico City, Mexico
José Emanuel Rodríguez-Fitta
Facultad de Estudios Superiores Acatlán, Universidad Nacional Autónoma de México
(UNAM), Av. Alcanfores y San Juan Totoltepec s/n, Santa Cruz Acatlán, Naucalpan,
53150, Edo. de México, Mexico
Prejeto 17. decembra 2020, sprejeto 2. maja 2022, objavljeno na spletu 11. novembra 2022
Povzetek
Preučujemo (v, k, λ)-simetrične dizajne, ki imajo praporno tranzitivno in točkovno
primitivno grupo avtomorfizmov, kjer je v = m2 in (k, λ) = t > 1, ter dokažemo, da
če je D dizajn s sodim m, ki dopušča praporno tranzitivno in točkovno primitivno grupo
avtomorfizmov G, potem velja bodisi:
(1) D je dizajn s parametri
(
(2t+ s− 1)2, 2t
2−(2−s)t
s ,
t2−t
s2
)
, kjer je s ≥ 1 lih, bodisi
(2) G nima netrivialnega produktnega delovanja.
Opazimo, da parametri v (1), kadar je s = 1, ustrezajo Menonovim dizajnom.
Dokažemo tudi, da če je D (v, k, λ)-simetričen dizajn s praporno tranzitivno in točkovno
primitivno grupo avtomorfizmov s produktnim delovanjem, kjer je v = ml in l ≥ 2, potem
komplement dizajna D ne dopušča praporno tranzitivne grupe avtomorfizov.
Ključne besede: Simetrični dizajni, praporna tranzitivnost, primitivne grupe, grupa avtomorfizmov
dizajnov.
*Avtorja bi rada izrazila svojo hvaležnost recenzentu za zelo koristne pripombe in predloge, ki so izboljšali
najin članek.
†Kontaktni avtor.
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/
Math. Subj. Class. (2020): 05B05, 51E05, 20B15, 20B25
E-poštna naslova: eugenia@im.unam.mx (Eugenia O’Reilly-Regueiro), 887191@pcpuma.acatlan.unam.mx
(José Emanuel Rodríguez-Fitta)