On the split structure of lifted groups

Malnič, Aleksander; Požar, Rok

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Title:	On the split structure of lifted groups
Authors:	ID Malnič, Aleksander (Author) ID Požar, Rok (Author)
Files:	RAZ_Malnic_Aleksander_i2016.pdf (422,56 KB) MD5: F55E1183DD33B056559D63221616BD55
Language:	English
Work type:	Unknown
Typology:	1.01 - Original Scientific Article
Organization:	ZUP - University of Primorska Press
Abstract:	Let $\wp \colon \tilde{X} \to X$ be a regular covering projection of connected graphs with the group of covering transformations $\rm{CT}_\wp$ being abelian. Assuming that a group of automorphisms $G \le \rm{Aut} X$ lifts along $\wp$ to a group $\tilde{G} \le \rm{Aut} \tilde{X}$ , the problem whether the corresponding exact sequence $\rm{id} \to \rm{CT}_\wp \to \tilde{G} \to G \to \rm{id}$ splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence. In the above combinatorial setting the extension is given only implicitly: neither $\tilde{G}$ nor the action $G\to \rm{Aut} \rm{CT}_\wp$ nor a 2-cocycle $G \times G \to \rm{CT}_\wp$ , are given. Explicitly constructing the cover $\tilde{X}$ together with $\rm{CT}_\wp$ and $\tilde{G}$ as permutation groups on $\tilde{X}$ is time and space consuming whenever $\rm{CT}_\wp$ is large; thus, using the implemented algorithms (for instance, HasComplement in Magma) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group); one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever $\rm{CT}_\wp$ is elementary abelian.
Keywords:	algorithm, abelian cover, Cayley voltages, covering projection, graph, group extension, group presentation, lifting automorphisms, linear systems over the integers, semidirect product
Year of publishing:	2016
Number of pages:	str. 113-134
Numbering:	Vol. 10, no. 1
PID:	20.500.12556/RUP-7200
UDC:	519.17
ISSN on article:	1855-3966
COBISS.SI-ID:	1537674948
Publication date in RUP:	15.10.2015
Views:	3575
Downloads:	167
Metadata:
:	MALNIČ, Aleksander and POŽAR, Rok, 2016, On the split structure of lifted groups. Ars mathematica contemporanea [online]. 2016. Vol. 10, no. 1, p. 113–134. [Accessed 21 April 2025]. Retrieved from: https://repozitorij.upr.si/IzpisGradiva.php?lang=eng&id=7200 Copy citation

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Title:	Ars mathematica contemporanea
Publisher:	Društvo matematikov, fizikov in astronomov, Društvo matematikov, fizikov in astronomov, Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije
ISSN:	1855-3966
COBISS.SI-ID:	239049984

Secondary language

Language:

English

Abstract:

Naj bo

$\wp \colon \tilde{X} \to X$ regularna krovna projekcija povezanih grafov, grupa krovnih transformacij

$\rm{CT}_\wp$ pa naj bo abelova. Ob predpostavki, da se grupa avtomorfizmov

$G \le \rm{Aut} X$ dvigne vzdolž

$\wp$ do grupe

$\tilde{G} \le \rm{Aut} \tilde{X}$ , podrobno analiziramo problem, ali se ustrezno eksaktno zaporedje

$\rm{id} \to \rm{CT}_\wp \to \tilde{G} \to G \to \rm{id}$ razcepi glede na Cayleyevo dodelitev napetosti, ki rekonstruira projekcijo do ekvivalence natančno. V gornjem kombinatoričnem sestavu je razširitev podana samo implicitno: podani niso ne

$\tilde{G}$ ne delovanje

$G\to \rm{Aut} \rm{CT}_\wp$ ne 2-kocikel

$G \times G \to \rm{CT}_\wp$ . Eksplicitno konstruiranje krova

$\tilde{X}$ ter

$\rm{CT}_\wp$ in

$\tilde{G}$ kot permutacijskih grup na

$\tilde{X}$ je časovno in prostorsko zahtevno vselej, kadar je

$\rm{CT}_\wp$ velik; tako je uporaba implementiranih algoritmov (na primer, HasComplement v Magmi) vse prej kot optimalna. Namesto tega pokažemo, da lahko najnujnejšo informacijo o delovanju in 2-kociklu učinkovito izluščimo neposredno iz napetosti (ne da bi eksplicitno konstruirali krov in dvignjeno grupo); zdaj bi bilo mogoče uporabiti standardno metodo reduciranja problema na reševanje sistema linearnih enačb nad celimi števili. Vendar tukaj uberemo malce drugačen pristop, ki sploh ne zahteva nobenega poznavanja kohomologije. Časovno in prostorsko zahtevnost formalno analiziramo za vse primere, ko je

$\rm{CT}_\wp$ elementarna abelova.

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