Length function

In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.

Definition

A length function L : G → R+ on a group G is a function satisfying:[1][2][3]

Compare with the axioms for a metric and a filtered algebra.

Word metric

Main article: Word metric

An important example of a length is the word metric: given a presentation of a group by generators and relations, the length of an element is the length of the shortest word expressing it.

Coxeter groups (including the symmetric group) have combinatorial important length functions, using the simple reflections as generators (thus each simple reflection has length 1). See also: length of a Weyl group element.

A longest element of a Coxeter group is both important and unique up to conjugation (up to different choice of simple reflections).

Properties

A group with a length function does not form a filtered group, meaning that the sublevel sets do not form subgroups in general.

However, the group algebra of a group with a length functions forms a filtered algebra: the axiom corresponds to the filtration axiom.

References

  1. ^ Lyndon, Roger C. (1963), "Length functions in groups", Mathematica Scandinavica, 12: 209–234, doi:10.7146/math.scand.a-10684, JSTOR 24489388, MR 0163947
  2. ^ Harrison, Nancy (1972), "Real length functions in groups", Transactions of the American Mathematical Society, 174: 77–106, doi:10.2307/1996098, MR 0308283
  3. ^ Chiswell, I. M. (1976), "Abstract length functions in groups", Mathematical Proceedings of the Cambridge Philosophical Society, 80 (3): 451–463, doi:10.1017/S0305004100053093, MR 0427480

This article incorporates material from Length function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Read other articles:

Rai 2

Rai 2Diluncurkan4 November 1961; 62 tahun lalu (1961-11-04)PemilikRaiPangsa pemirsa6.85% (Mei 2014, [1])NegaraItaliaBahasaItaliaKantor pusatRoma, ItaiaSaluran seindukRai 1Rai 3Rai 4Rai 5Rai GulpRai MovieRai News 24Rai PremiumRai ScuolaRai SportRai StoriaRai YoyoRai LadiniaRai SüdtirolRai ItaliaSitus webrai.it/rai2Televisi InternetRai.tvLive Streaming Rai 2 (bahasa Italia: Rai Due) adalah sebuah saluran televisi publik kedua yang disiarkan oleh penyiar milik negara Italia, RA...

 

Anoplophora leechi

Anoplophora leechi Klasifikasi ilmiah Kerajaan: Animalia Filum: Arthropoda Kelas: Insecta Ordo: Coleoptera Famili: Cerambycidae Subfamili: Lamiinae Tribus: Lamiini Genus: Anoplophora Spesies: Anoplophora leechi Anoplophora leechi adalah spesies kumbang tanduk panjang yang tergolong familia Cerambycidae. Spesies ini juga merupakan bagian dari genus Anoplophora, ordo Coleoptera, kelas Insecta, filum Arthropoda, dan kingdom Animalia. Larva kumbang ini biasanya mengebor ke dalam kayu dan dapat m...

 

Fly Me to the Moon (manga)

Halaman ini berisi artikel tentang seri manga. Untuk kegunaan lain, lihat Fly Me to the Moon (disambiguasi). Fly Me to the MoonGambar sampul manga volume pertamaトニカクカワイイ(Tonikaku Kawaii)GenreIyashikei MangaPengarangKenjiro Hata (畑健二郎)PenerbitShogakukanPenerbit bahasa IndonesiaElex Media KomputindoImprintShōnen Sunday ComicsMajalahWeekly Shōnen SundayDemografiShōnenTerbit14 Februari 2018 – sekarangVolume26 (Daftar volume) Seri animeSutradaraHiroshi Ikehata (博史�...

Ankole

Kerajaan Ankole dan distriknya Bendera Kerajaan Ankole Ankole, juga disebut Nkore, adalah salah satu dari empat kerajaan tradisional di Uganda. Kerajaan ini terletak di barat daya Uganda, sebelah timur Danau Edward. Kerajaan ini dikuasai oleh Raja yang disebut Mugabe atau Omugabe dari Ankole. Kerajaan ini secara resmi dibubarkan tahun 1967 oleh pemerintahan Presiden Milton Obote, dan secara resmi masih belum direstorasi.[1] Penduduk Ankole disebut Banyankole (singular: Munyankole) dal...

 

Aek Ledong, Aek Ledong, Asahan

Aek LedongDesa Kantor Kepala Desa Aek LedongNegara IndonesiaProvinsiSumatera UtaraKabupatenAsahanKecamatanAek LedongKode pos21273Kode Kemendagri12.09.32.2003 Luas... km2Jumlah penduduk548 jiwa[1]Kepadatan... jiwa/km2 Gapura selamat datang di Desa Aek Ledong Aek Ledong merupakan salah satu desa yang ada di kecamatan Aek Ledong, Kabupaten Asahan, provinsi Sumatera Utara, Indonesia. Referensi ^ (Inggris) Sensus penduduk tahun 2010, halaman 216 lbsKecamatan Aek Ledong, Kabupaten Asah...

 

KDDI

KDDIIndustrijasa telekomunikasiDidirikan1984KantorpusatJepangWilayah operasiSeluruh duniaPendapatan$42,3 miliarTotal aset$46,1 miliarSitus webwww.kddi.com KDDI adalah sebuah perusahaan asal Jepang yang bergerak di sektor jasa telekomunikasi.[1] Industri yang menjadi fokus utama KDDI adalah industri jasa telekomunikasi.[1] Pada tahun 2014, KDDI mendapatkan nilai penjualan sebesar AS$42,3 miliar dengan profit AS$3,4 miliar.[1] Pada tahun yang sama, KDDI menempati peringk...

Lys Mousset

French footballer (born 1996) Lys Mousset Mousset in 2018Personal informationFull name Lys Émilien Mousset[1]Date of birth (1996-02-08) 8 February 1996 (age 28)[2]Place of birth Montivilliers, FranceHeight 1.84 m (6 ft 0 in)[2]Position(s) ForwardYouth career2003–2004 Soquence Graville2004–2006 Havre Caucriauville2006–2012 Le HavreSenior career*Years Team Apps (Gls)2012–2015 Le Havre B 57 (14)2014–2016 Le Havre 34 (14)2016–2019 AFC Bourn...

 

Orientation (Agents of S.H.I.E.L.D.)

1st and 2nd episodes of the 5th season of Agents of S.H.I.E.L.D. OrientationAgents of S.H.I.E.L.D. episodesThis episode begins the first pod of episodes with a new typeface logo for the series.[1]Episode nos.Season 5Episodes 1 and 2Directed by Jesse Bochco David Solomon Written by Jed Whedon Maurissa Tancharoen DJ Doyle Produced by Jed Whedon Maurissa Tancharoen Jeffrey Bell Cinematography byAllan WestbrookEditing byDexter AdrianoOriginal air dateDecember 1, 2017 ...

 

Население Татарстана

Численность населения республики по данным Росстата составляет 4 003 016[1] чел. (2024). Татарстан занимает 8-е место по численности населения среди субъектов Российской Федерации[2]. Плотность населения — 59,00 чел./км² (2024). Городское население — 76,72[3] % (20...

List of unsolved problems in physics

This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources. The following is a list of notable unsolved problems grouped into broad areas of physics.[1] Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result. The others are experimental, meaning that there is a difficult...

 

Філософія

Частина серії проФілософіяLeft to right: Plato, Kant, Nietzsche, Buddha, Confucius, AverroesПлатонКантНіцшеБуддаКонфуційАверроес Філософи Епістемологи Естетики Етики Логіки Метафізики Соціально-політичні філософи Традиції Аналітична Арістотелівська Африканська Близькосхідна іранська Буддій�...

 

Roxsolt Liv SRAM

Australian cycling team Roxsolt Liv SRAMTeam informationUCI codeROX (2013–2019)RXS (2020–)RegisteredAustraliaFounded2013 (2013)StatusNational (2013–2019)UCI Women's Continental Team (2020–)Team name history2013–20182019–20202021–RoxsoltRoxsolt Attaquer[1]Roxsolt Liv SRAM Roxsolt Liv SRAM is an Australian women's road bicycle racing team which participates in elite women's races.[2][3] The team was established in 2013. Team roster As of 3 February 202...

Arup Raha

Air Chief MarshalArup RahaPVSM, AVSM, VM, ADCঅরূপ রাহাOfficial portrait, 201454th Chairman of the Chiefs of Staff CommitteeIn office31 July 2014 – 31 December 2016PresidentPranab MukherjeePrime MinisterNarendra ModiPreceded byBikram SinghSucceeded bySunil Lanba21st Chief of Air StaffIn office31 December 2013 – 31 December 2016PresidentPranab MukherjeePrime MinisterManmohan SinghNarendra ModiPreceded byNorman Anil Kumar BrowneSucceeded byBirender Singh Dh...

 

Anti-Korean sentiment in Japan

See also: Anti-Japanese sentiment in Korea This article may be unbalanced toward certain viewpoints. Please improve the article by adding information on neglected viewpoints, or discuss the issue on the talk page. (May 2023) Part of a series onKoreans in Japan History Korea under Japanese rule Korean independence movement Hanshin Education Incident Culture Chōsen-seki South Korean nationality law Special permanent resident (Japan) Zainichi Korean language Enclaves Ikuno Korea Town Ōkubo, T...

 

Jawa Timur VII (daerah pemilihan)

Jawa Timur VIIDaerah pemilihan Daerah Pemilihan /untuk Dewan Perwakilan RakyatRepublik IndonesiaWilayah Daftar Kabupaten : Magetan Ngawi Pacitan Ponorogo Trenggalek ProvinsiJawa TimurPopulasi3.903.963 (2023)[1]Elektorat3.059.436 (2024)[2]Daerah pemilihan saat iniDibentuk2004Kursi8Anggota  Ibnu Multazam (PKB)  Supriyanto (Gerindra)  Ina Ammania (PDI-P)  Johan Budi (PDI-P)  Ali Mufthi (Golkar)  Sri Wahyuni (NasDem)  Edhie Baskoro Yudhoyono...

الناقورة (لبنان)

  هذه المقالة عن الناقورة القرية اللبنانية. لمعانٍ أخرى، طالع الناقورة (توضيح). الناقورة   الاسم الرسمي الناقورة  الإحداثيات 33°07′12″N 35°08′03″E / 33.12°N 35.134166666667°E / 33.12; 35.134166666667   تقسيم إداري  البلد لبنان[1]  التقسيم الأعلى قضاء صور  خصائص جغر...

 

Mount Zwischen

Mountain in the state of Colorado Mount ZwischenEast aspectHighest pointElevation12,011 ft (3,661 m)[1][2]Prominence2,266 ft (691 m)[2]Isolation4.44 mi (7.15 km)[2]ListingColorado prominent summitsCoordinates37°47′28″N 105°27′19″W / 37.7912185°N 105.4554033°W / 37.7912185; -105.4554033[3]GeographyMount ZwischenColorado LocationHuerfano and Saguache counties, Colorado, United States&#...

 

Sakura, Saku

Japanese manga series Sakura, SakuFirst tankōbon volume coverサクラ、サク。GenreRomance[1] MangaWritten byIo SakisakaPublished byShueishaEnglish publisherNA: Viz MediaImprintMargaret ComicsMagazineBessatsu MargaretDemographicShōjoOriginal runFebruary 13, 2021 – October 13, 2023Volumes9 (List of volumes) Sakura, Saku (サクラ、サク。) is a Japanese manga series written and illustrated by Io Sakisaka. It was serialized in Shueisha's Bessatsu Margaret magazine ...

Cariri Velhos belt

Belt of rocks in Brazil The Cariri Velhos belt is a belt of rocks in Brazil that were deformed and metamorphosed in the Early Neoproterozoic.[1] The belt runs from the Atlantic more than 700 kilometres (430 mi) inland in a SWW direction. The width of the belt varies from 100 to 50 kilometres (62 to 31 mi).[2] The Cariri Velhos belt lies intercalated in a mosaic with various Brasiliano cycle belts in the Borborema Province, a geologic province in Northeastern Brazil. ...

 

Two Arabian Knights

Two Arabian KnightsSutradaraLewis MilestoneProduserJohn W. Considine Jr.Jeffery MasinoHoward HughesDitulis olehDonald McGibneyWallace SmithCyril GardnerJames T. O'DonohoeGeorge Marion Jr.PemeranWilliam BoydMary AstorLouis WolheimSinematograferTony GaudioJoseph H. AugustPenyuntingDouglass BiggsDistributorUnited ArtistsTanggal rilis 23 September 1927 (1927-09-23) Durasi92 menitNegaraAmerika SerikatBahasaAntarjudul Inggris Two Arabian Knights (1927) adalah sebuah film komedi Amerika, yang d...