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 QuestionWhat 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 DebateWhether 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 Connections
Associated With
Includes
Source E8 Lattice (Wikipedia)
Source Eugenio Elia Levi (Wikipedia)
Source Group Theory (Wikipedia)
Source Ferdinand Georg Frobenius (Wikipedia)
Source Free Product (Wikipedia)
Source Grigorchuk Group (Wikipedia)
Source Group Theory (Wikipedia)
Source Heinrich Tietze (Wikipedia)
Source Higman's embedding theorem (Wikipedia)
Source Lagrange's Theorem, Group Theory (Wikipedia)
Source Leech Lattice (Wikipedia)
Source Mathieu group M11 (Wikipedia)
Source Monster Group (Wikipedia)
Source Rank of a Group (Wikipedia)
Sources
1. Group Theory (Wikipedia)
Wikipedialead 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: Evariste Galois, History section
Evariste Galois coined the term group and established a connection, now known as Galois theory, between the nascent theory of groups and field theory.
View the Source 2. Classification of Finite Simple Groups (Wikipedia)
WikipediaSecond-generation classification sectionQuote, 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.
Lagrange's Theorem, Group Theory (Wikipedia)
Wikimedia FoundationIncludes: Lagrange's Theorem (Group Theory)View the Source Monster Group (Wikipedia)
E8 Lattice (Wikipedia)
Leech Lattice (Wikipedia)
Mathieu group M11 (Wikipedia)
Includes: Mathieu Group M11, Lead sentenceQuote, 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
Free Product (Wikipedia)
Includes: Free Product, Lead sentenceQuote, 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
Rank of a Group (Wikipedia)
Includes: Rank of a Group, Lead sentenceQuote, 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
Higman's embedding theorem (Wikipedia)
Includes: Higman's Embedding Theorem, Lead sentenceQuote, 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
Grigorchuk Group (Wikipedia)
Includes: Grigorchuk Group, Lead sentenceQuote, 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
Ferdinand Georg Frobenius (Wikipedia)
Includes: Ferdinand Georg Frobenius, Lead paragraph [in-branch]Quote, Includes: Ferdinand Georg Frobenius, Lead paragraph [in-branch]
group theory
Heinrich Tietze (Wikipedia)
Includes: Heinrich Tietze, Lead paragraph [in-branch 2]Quote, Includes: Heinrich Tietze, Lead paragraph [in-branch 2]
group presentations
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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