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Why Do Sporadic Groups Exist?

⬣ LINKS ⬣ ⬡ PATREON:   / anotherroof   ⬡ CHANNEL:    / anotherroof   ⬡ WEBSITE: https://anotherroof.top ⬡ SUBREDDIT:   / anotherroof   ⬡ TWITCH:   / anotherroof   Permutation Group Visualiser by Tesseralis: https://permutation-groups.glitch.me/ ⬣ ABOUT ⬣ The concept of symmetry can be formalised through the study of groups. Groups can be constructed from simple groups, and during an incredible effort throughout the 20th century, all simple groups have been classified. But the classification is messy -- there are several infinite families and then 26 left-over groups which don't fit any of the patterns. In this video, we explore some of these so-called Sporadic Groups and why they exist. ⬣ TIMESTAMPS ⬣ 00:00 - Intro and Housekeeping 01:56 - Introduction to Group Theory 05:13 - Classifying Groups 10:04 - Transitive Groups 15:06 - M24 19:20 - The Mathieu Family 26:29 - Why Sporadic Groups Exist ⬣ INVESTIGATORS ⬣ Nothing for you here. Sorry! ⬣ REFERENCES ⬣ [1] É Mathieu, Sur la fonction cinq fois transitive de 24 quantitiés. Journal de mathématiques pures et appliquées 18 (1873) pp. 25-46. [2] E Witt, Die 5-fach transitiven Gruppen von Mathieu. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 12: 256–264 [3] P Cameron, Projective and polar spaces. (1992) University of London, Queen Mary and Westfield College. [4] P Cameron, Permtation groups. (1999) London Mathematical Society Student Texts, Cambridge University Press. [5] W Burnside, Theory of Groups of Finite Order. (1911) Cambridge University P [6] https://mathshistory.st-andrews.ac.uk... [7] P Duhem, Emile Mathieu, his life and works. Bull. New York Math. Soc. 1(7): 156-168 (April 1892). [8] Aimo Tietäväinen, On the nonexistence of perfect codes over finite fields. SIAM Journal on Applied Mathematics, 24(1), 88–96. ⬣ CREDITS ⬣ Music by Danjel Zambo. Intro music by Tobias Voigt. Image Credits Molecule https://commons.wikimedia.org/wiki/Fi... Periodic table https://upload.wikimedia.org/wikipedi... Tesseract https://en.wikipedia.org/wiki/File:Te... Rings https://media.distractify.com/brand-i... Rack https://upload.wikimedia.org/wikipedi... Magma https://media.wired.co.uk/photos/606d... Lattice https://static.buildeazy.com/wp-conte... Knot (quandle) https://math.illinois.edu/sites/defau... Ernst Witt https://en.wikipedia.org/wiki/Ernst_W... Bernd Fischer https://upload.wikimedia.org/wikipedi... Robert Griess https://upload.wikimedia.org/wikipedi... Fisher-Price Monster https://www.coconuttoysandfancydress.... E8 Diagram https://upload.wikimedia.org/wikipedi... Circular arrows https://www.freepik.com/free-vector/r... G2 Diagram https://en.wikipedia.org/wiki/G2_(mat... F4 Diagram https://en.wikipedia.org/wiki/F4_(mat... E6 Diagram https://en.wikipedia.org/wiki/E6_(mat... E7 Diagram https://en.wikipedia.org/wiki/E7_(mat... William Burnside https://upload.wikimedia.org/wikipedi... Platonic Solids https://commons.wikimedia.org/wiki/Fi... Explosion https://www.vecteezy.com/video/297431... Explosion    • Explosion croma key green screen, wit...   Saturn, Jupiter timelapse, Voyager 1 NASA Public Domain Image