ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P1.09
https://doi.org/10.26493/1855-3974.2975.1b2
(Also available at http://amc-journal.eu)
Products of subgroups, subnormality,
and relative orders of elements
Luca Sabatini *
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
Reáltanoda utca 13-15, H-1053, Budapest, Hungary
Received 6 October 2022, accepted 22 February 2023, published online 31 August 2023
Abstract
Let G be a group. We give an explicit description of the set of elements x ∈ G such
that x|G:H| ∈ H for every subgroup of finite index H ⩽ G. This is related to the following
problem: given two subgroups H and K, with H of finite index, when does |HK : H|
divide |G : H|?
Keywords: Relative order, product of subgroups, subnormal subgroup.
Math. Subj. Class. (2020): 20D40, 20D25, 20F99.
*The author thanks Bob Guralnick and Orazio Puglisi for useful conversations.
E-mail address: sabatini.math@gmail.com (Luca Sabatini)
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 24 (2024) #P1.09
https://doi.org/10.26493/1855-3974.2975.1b2
(Dostopno tudi na http://amc-journal.eu)
Produkti podgrup, subnormalnost
in relativni vrstni red elementov
Luca Sabatini *
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
Reáltanoda utca 13-15, H-1053, Budapest, Hungary
Prejeto 6. oktobra 2022, sprejeto 22. februarja 2023, objavljeno na spletu 31. avgusta 2023
Povzetek
Naj bo G grupa. Podamo ekspliciten opis množice elementov x ∈ G, za katere je
x|G:H| ∈ H za vsako podgrupo končnega indeksa H ⩽ G. To je povezano z naslednjim
problemom: če sta dani dve podgrupi H in K, in ima H končen indeks, kdaj |HK : H|
deli |G : H|?
Ključne besede: Relativni vrstni red, produkt podgrup, podnormalna podgrupa.
Math. Subj. Class. (2020): 20D40, 20D25, 20F99.
*Avtor se zahvaljuje Bobu Guralnicku in Oraziu Puglisiju za koristne pogovore.
E-poštni naslov: sabatini.math@gmail.com (Luca Sabatini)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/