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Group Theory

Algebra

Group theory is the branch of abstract algebra that studies groups, algebraic structures consisting of a set together with an operation that combines any two elements while satisfying closure, associativity, an identity element and invertibility. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Central Question
What are the finite simple groups, the basic building blocks every finite group is assembled from, and can a complete list of them be proven. 1
Key Debate
Whether the classification of finite simple groups, a proof spread across tens of thousands of journal pages by hundreds of authors, can be trusted as complete. A 1983 announcement that the work was finished proved premature once a gap in the quasithin case came to light, and a second-generation proof project has since worked toward a shorter, more checkable version. 2
Classification
Pure or Applied
Pure Mathematics 1
Connections

Associated With

Includes

Source MacTutor biography of William Burnside, St Andrews

Showing 51 to 51 of 51 in Includes, page 2 of 2.

Sources
1. Group Theory (Wikipedia)
Wikipedia
  • lead paragraph
    In abstract algebra, group theory studies the algebraic structures known as groups.
  • opening section
    One of the most important mathematical achievements of the 20th century was the collaborative effort ... that culminated into a complete classification of finite simple groups.
  • Includes: Évariste Galois, History
    Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory.
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2. Classification of Finite Simple Groups (Wikipedia)
WikipediaSecond-generation classification section
Quote, Second-generation classification section
Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups.
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Lagrange's Theorem, Group Theory (Wikipedia)
Wikimedia FoundationIncludes: Lagrange's Theorem (Group Theory)View the Source
Monster Group (Wikipedia)
Wikimedia FoundationIncludes: Monster GroupView the Source
E8 Lattice (Wikipedia)
Wikimedia FoundationIncludes: E8 LatticeView the Source
Leech Lattice (Wikipedia)
Wikimedia FoundationIncludes: Leech LatticeView the Source
Wikipedia: Modular group
Includes: Modular Group, Lead
Quote, Includes: Modular Group, Lead
the modular group is the projective special linear group
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Marcel Herzog (Mathematics Genealogy Project)
Includes: Marcel Herzog, Degree record
Quote, Includes: Marcel Herzog, Degree record
On Finite Groups with a Subgroup Containing the Normalizer of Each of Its Nonunit Elements
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Mathieu group M11 (Wikipedia)
Includes: Mathieu Group M11, Lead sentence
Quote, Includes: Mathieu Group M11, Lead sentence
In the area of modern algebra known as group theory, the Mathieu group M11 is a sporadic simple group of order 7,920 = 11 · 10 · 9
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Free Product (Wikipedia)
Includes: Free Product, Lead sentence
Quote, Includes: Free Product, Lead sentence
In mathematics, specifically group theory, the free product is an operation that takes two groups ⁠ G ⁠ and ⁠ H ⁠ and constructs a
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Rank of a Group (Wikipedia)
Includes: Rank of a Group, Lead sentence
Quote, Includes: Rank of a Group, Lead sentence
In the mathematical subject of group theory, the rank of a group G, denoted rank(G), can refer to the smallest cardinality of a ge
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Higman's embedding theorem (Wikipedia)
Includes: Higman's Embedding Theorem, Lead sentence
Quote, Includes: Higman's Embedding Theorem, Lead sentence
In group theory, Higman's embedding theorem states that every finitely generated recursively presented group R can be embedded as
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Grigorchuk Group (Wikipedia)
Includes: Grigorchuk Group, Lead sentence
Quote, Includes: Grigorchuk Group, Lead sentence
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constru
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Ferdinand Georg Frobenius (Wikipedia)
Includes: Ferdinand Georg Frobenius, Lead paragraph [in-branch]
Quote, Includes: Ferdinand Georg Frobenius, Lead paragraph [in-branch]
group theory
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Heinrich Tietze (Wikipedia)
Includes: Heinrich Tietze, Lead paragraph [in-branch 2]
Quote, Includes: Heinrich Tietze, Lead paragraph [in-branch 2]
group presentations
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Eugenio Elia Levi (Wikipedia)
Includes: Eugenio Elia Levi, Lead paragraph [in-branch]
Quote, Includes: Eugenio Elia Levi, Lead paragraph [in-branch]
group theory
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Wikipedia: Alfred Young
Includes: Alfred Young, Lead
Quote, Includes: Alfred Young, Lead
He is known for his work in the area of group theory.
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Wikipedia: Rostislav Grigorchuk
Includes: Rostislav Grigorchuk, Lead
Quote, Includes: Rostislav Grigorchuk, Lead
working in different areas of mathematics including group theory
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Wikipedia: Philip Hall
Includes: Philip Hall, Lead
Quote, Includes: Philip Hall, Lead
His major work was on group theory, notably on finite groups and solvable groups.
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Wikipedia: Walter Feit
Includes: Walter Feit, Lead
Quote, Includes: Walter Feit, Lead
Walter Feit (October 26, 1930, July 29, 2004) was an Austrian-born American mathematician who worked in finite group theory and representation theory.
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Wikipedia: John G. Thompson
Includes: John G. Thompson, Lead
Quote, Includes: John G. Thompson, Lead
John Griggs Thompson (born October 13, 1932) is an American mathematician at the University of Florida noted for his work in the field of finite groups.
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Wikipedia: Mikhael Gromov (mathematician)
Includes: Mikhail Gromov, Lead
Quote, Includes: Mikhail Gromov, Lead
Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; Russian: Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory.
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Wikipedia: Graham Higman
Includes: Graham Higman, Lead
Quote, Includes: Graham Higman, Lead
Graham Higman FRS (19 January 1917-8 April 2008) was a prominent English mathematician known for his contributions to group theory. == Biography == Higman was born in Louth, Lincolnshire, and attended Sutton High School, Plymouth, winning a scholarship to Balliol College, Oxford.
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Wikipedia: Max Dehn
Includes: Max Dehn, Lead
Quote, Includes: Max Dehn, Lead
Max Wilhelm Dehn (November 13, 1878, June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory.
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Camille Jordan (Wikipedia)
Wikimedia FoundationIncludes: Camille Jordan, Lead paragraph
Quote, Includes: Camille Jordan, Lead paragraph
Marie Ennemond Camille Jordan (French: [ʒɔʁdɑ̃]; 5 January 1838-22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his textbook Cours d'analyse de l'École polytechnique.
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Herzog-Schönheim conjecture (Wikipedia)
Includes: Herzog-Schonheim Conjecture, Lead paragraph
Quote, Includes: Herzog-Schonheim Conjecture, Lead paragraph
In mathematics, the Herzog-Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974.
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Wikipedia: Dihedral group
Includes: Dihedral Group, Lead paragraph
Quote, Includes: Dihedral Group, Lead paragraph
Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
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Wikipedia: Cayley graph
Includes: Cayley Graph, Lead paragraph
Quote, Includes: Cayley Graph, Lead paragraph
It is a central tool in combinatorial and geometric group theory.
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Newton's identities, Wikipedia
Includes: Newton's Identities, Lead paragraph
Quote, Includes: Newton's Identities, Lead paragraph
They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.
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Heinrich Maschke, Biography (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsIncludes: Heinrich Maschke, Biography
Quote, Includes: Heinrich Maschke, Biography
who worked on group theory and on differential geometry
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Wolfram MathWorld: Alternating Group
Wolfram MathWorldIncludes: Alternating Group, Subject classifications
Quote, Includes: Alternating Group, Subject classifications
Algebra Group Theory
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Wolfram MathWorld: Heisenberg Group
Wolfram MathWorldIncludes: Heisenberg Group, Subject classifications
Quote, Includes: Heisenberg Group, Subject classifications
Algebra Group Theory
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Wolfram MathWorld: Additive Inverse
Wolfram MathWorldIncludes: Additive Inverse, Subject classifications
Quote, Includes: Additive Inverse, Subject classifications
Algebra Group Theory
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Wolfram MathWorld: General Linear Group
Wolfram MathWorldIncludes: General Linear Group, Subject classifications
Quote, Includes: General Linear Group, Subject classifications
Algebra Group Theory
View the Source
Wolfram MathWorld: Quaternion Group
Wolfram MathWorldIncludes: Quaternion Group, Subject classifications
Quote, Includes: Quaternion Group, Subject classifications
Algebra Group Theory
View the Source
Wolfram MathWorld: Free Group
Wolfram MathWorldIncludes: Free Group, Subject classifications
Quote, Includes: Free Group, Subject classifications
Algebra Group Theory
View the Source
Wolfram MathWorld: Coxeter Element
Wolfram MathWorldIncludes: Coxeter Element, Subject classifications
Quote, Includes: Coxeter Element, Subject classifications
Algebra Group Theory
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Andrews-Curtis conjecture (Wikipedia)
Includes: Andrews-Curtis Conjecture, Lead section
Quote, Includes: Andrews-Curtis Conjecture, Lead section
the Andrews-Curtis conjecture states that every balanced presentation (i.e. a presentation with the same number of
View the Source
Wikipedia: Klein four-group
Includes: Klein Four-Group, Lead section
Quote, Includes: Klein Four-Group, Lead section
In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing
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Rubik's Cube group (Wikipedia)
Includes: Rubik's Cube Group, Lead section
Quote, Includes: Rubik's Cube Group, Lead section
Rubik's Cube group
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Wikipedia: Krull-Schmidt theorem
Wikimedia FoundationIncludes: Krull-Schmidt Theorem, Lead section
Quote, Includes: Krull-Schmidt Theorem, Lead section
that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product
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MacTutor biography of Felix Klein, St Andrews
Includes: Felix Klein, Biography
Quote, Includes: Felix Klein, Biography
his work on the connections between geometry and group theory
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MacTutor biography of John Leech, St Andrews
Includes: John Leech, Summary
Quote, Includes: John Leech, Summary
John Leech was an English mathematician who is best known for the Leech lattice which is important in the theory of finite simple groups.
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MacTutor biography of Ludwig Sylow, St Andrews
Includes: Ludwig Sylow, Summary
Quote, Includes: Ludwig Sylow, Summary
Ludwig Sylow was a high school teacher who proved what is perhaps the most profound result in the theory of finite groups.
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MacTutor biography of William Burnside, St Andrews
Includes: William Burnside, Summary
Quote, Includes: William Burnside, Summary
William Burnside wrote the first treatise on groups in English and was the first to develop the theory of groups from a modern abstract point of view.
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Otto Schreier, Biography (MacTutor History of Mathematics)
Includes: Otto Schreier, Summary
Quote, Includes: Otto Schreier, Summary
Otto Schreier worked in combinatorial group theory, particularly on subgroups of free groups and on knot groups.
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Reinhold Baer, Biography (MacTutor History of Mathematics)
Includes: Reinhold Baer, Summary
Quote, Includes: Reinhold Baer, Summary
Reinhold Baer 's mathematical work was wide ranging; topology, abelian groups and geometry. His most important work, however, was in group theory, on the extension problem for groups, finiteness conditions, soluble and nilpotent groups.
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Walther von Dyck, Biography (MacTutor History of Mathematics)
Includes: Walther von Dyck, Summary
Quote, Includes: Walther von Dyck, Summary
Von Dyck made important contributions to function theory, group theory ( where a fundamental result on group presentations is named after him ) topology and potential theory.
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Wilhelm Magnus, Biography (MacTutor History of Mathematics)
Includes: Wilhelm Magnus, Summary
Quote, Includes: Wilhelm Magnus, Summary
Wilhelm Magnus was a German American mathematician who worked in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations.
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Wilhelm Specht, Biography (MacTutor History of Mathematics)
Includes: Wilhelm Specht, Summary
Quote, Includes: Wilhelm Specht, Summary
Wilhelm Specht was a German mathematician who in worked group theory and representation theory.
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