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63 relations: Abelian group, Alperin–Brauer–Gorenstein theorem, Alternating group, American Mathematical Society, Augustin-Louis Cauchy, Burnside theorem, Cambridge University Press, Cauchy's theorem (group theory), Centralizer and normalizer, Classification of finite simple groups, Commutator subgroup, Computational group theory, Conjugacy class, Coprime integers, Coset, Cyclic group, Cyclic permutation, Dihedral group, Direct product, Equivalence class, Exponentiation, Finite group, Focal subgroup theorem, Frattini's argument, GAP (computer algebra system), Group action, Group isomorphism, Hall subgroup, Historia Mathematica, Index of a subgroup, Isomorphism, Jonathan Lazare Alperin, Lagrange's theorem (group theory), Lecture Notes in Computer Science, Magma (computer algebra system), Mathematician, Mathematics, Mathematische Annalen, Maximal subgroup, Multiplicity (mathematics), Normal p-complement, Normal subgroup, Norway, Order (group theory), Outer automorphism group, P-adic order, P-group, Permutation group, Peter Ludwig Mejdell Sylow, Prime number, ..., Prime power, Primitive root modulo n, Quasidihedral group, Semidirect product, Simple group, Solvable group, Springer Science+Business Media, Subgroup, Theorem, Time complexity, Up to, Wilson's theorem, Zorn's lemma. Expand index (13 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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Alperin–Brauer–Gorenstein theorem
In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2-subgroups.
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Alternating group
In mathematics, an alternating group is the group of even permutations of a finite set.
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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
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Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.
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Burnside theorem
In mathematics, Burnside theorem in group theory states that if G is a finite group of order where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Cauchy's theorem (group theory)
Cauchy's theorem is a theorem in the mathematics of group theory, named after Augustin Louis Cauchy.
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Centralizer and normalizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset S of a group G is the set of elements of G that commute with each element of S, and the normalizer of S are elements that satisfy a weaker condition.
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Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below.
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Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
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Computational group theory
In mathematics, computational group theory is the study of groups by means of computers.
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Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure.
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Coprime integers
In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.
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Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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Cyclic permutation
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of X. If S has k elements, the cycle is called a k-cycle.
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Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.
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Direct product
In mathematics, one can often define a direct product of objects already known, giving a new one.
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Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.
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Exponentiation
Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.
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Finite group
In abstract algebra, a finite group is a mathematical group with a finite number of elements.
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Focal subgroup theorem
In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group.
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Frattini's argument
In group theory, a branch of mathematics, Frattini's argument is an important lemma in the structure theory of finite groups.
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GAP (computer algebra system)
GAP (Groups, Algorithms and Programming) is a computer algebra system for computational discrete algebra with particular emphasis on computational group theory.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
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Hall subgroup
In mathematics, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index.
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Historia Mathematica
Historia Mathematica: International Journal of History of Mathematics is an academic journal on the history of mathematics published by Elsevier.
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Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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Jonathan Lazare Alperin
Jonathan Lazare Alperin (born 1937) is an American mathematician specializing in the area of algebra known as group theory.
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Lagrange's theorem (group theory)
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange.
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Lecture Notes in Computer Science
Springer Lecture Notes in Computer Science (LNCS) is a series of computer science books published by Springer Science+Business Media (formerly Springer-Verlag) since 1973.
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Magma (computer algebra system)
Magma is a computer algebra system designed to solve problems in algebra, number theory, geometry and combinatorics.
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Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Mathematische Annalen
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.
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Maximal subgroup
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.
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Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset.
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Normal p-complement
In mathematical group theory, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power of p. In other words the group is a semidirect product of the normal p-complement and any Sylow ''p''-subgroup.
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Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.
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Norway
Norway (Norwegian: (Bokmål) or (Nynorsk); Norga), officially the Kingdom of Norway, is a unitary sovereign state whose territory comprises the western portion of the Scandinavian Peninsula plus the remote island of Jan Mayen and the archipelago of Svalbard.
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Order (group theory)
In group theory, a branch of mathematics, the term order is used in two unrelated senses.
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Outer automorphism group
In mathematics, the outer automorphism group of a group,, is the quotient,, where is the automorphism group of and) is the subgroup consisting of inner automorphisms.
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P-adic order
In number theory, for a given prime number, the -adic order or -adic valuation of a non-zero integer is the highest exponent such that divides.
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P-group
In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order.
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Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself).
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Peter Ludwig Mejdell Sylow
Peter Ludwig Mejdell Sylow (12 December 1832 – 7 September 1918) was a Norwegian mathematician who proved foundational results in group theory.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Prime power
In mathematics, a prime power is a positive integer power of a single prime number.
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Primitive root modulo n
In modular arithmetic, a branch of number theory, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, for every integer a coprime to n, there is an integer k such that gk ≡ a (mod n).
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Quasidihedral group
In mathematics, the quasi-dihedral groups, also called semi-dihedral groups, are certain non-abelian groups of order a power of 2.
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Semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.
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Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.
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Solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Subgroup
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.
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Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms.
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Time complexity
In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.
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Up to
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
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Wilson's theorem
In number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), one has that the factorial (n - 1)!.
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Zorn's lemma
Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory that states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
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First Sylow Theorem, P subgroup, P-subgroup, Sylow 2-subgroup, Sylow Theorem, Sylow first theorem, Sylow group, Sylow p-subgroup, Sylow p-subgroups, Sylow subgroup, Sylow theorem, Sylow third theorem, Sylow's First theorem, Sylow's first theorem, Sylow's theorem, Sylow's theorems, Sylow's third theorem.
References
[1] https://en.wikipedia.org/wiki/Sylow_theorems
