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54 relations: Abelian group, Alternating group, Automorphism, Bit array, Bitwise operation, Cayley table, Cyclic group, Dihedral group, Dihedral group of order 6, Direct product of groups, Direct sum, Elementary abelian group, Empty set, Felix Klein, Field of sets, Finite ring, Galois theory, Graph (discrete mathematics), Group (mathematics), Group homomorphism, Group isomorphism, Identity component, Identity element, Kleinian group, List of small groups, Lodovico Ferrari, Mathematics, Milton Babbitt, Modular arithmetic, Multiplicative group of integers modulo n, Musical composition, Normal subgroup, Order (group theory), Oxford University Press, Positive real numbers, Power set, Presentation of a group, Quartic function, Quaternion group, Quotient group, Radical of an algebraic group, Rectangle, Representation theory of the symmetric group, Resolvent (Galois theory), Rhombus, Split-complex number, Springer Science+Business Media, Square, Subset, Symmetric difference, ..., Symmetric group, Symmetry group, Twelve-tone technique, Unit (ring theory). Expand index (4 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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Alternating group
In mathematics, an alternating group is the group of even permutations of a finite set.
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Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself.
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Bit array
A bit array (also known as bit map, bit set, bit string, or bit vector) is an array data structure that compactly stores bits.
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Bitwise operation
In digital computer programming, a bitwise operation operates on one or more bit patterns or binary numerals at the level of their individual bits.
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Cayley table
A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table.
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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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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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Dihedral group of order 6
In mathematics, the smallest non-abelian group has 6 elements.
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Direct product of groups
In group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted.
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Direct sum
The direct sum is an operation from abstract algebra, a branch of mathematics.
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Elementary abelian group
In group theory, an elementary abelian group (or elementary abelian p-group) is an abelian group in which every nontrivial element has order p. The number p must be prime, and the elementary abelian groups are a particular kind of ''p''-group.
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Empty set
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
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Felix Klein
Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory.
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Field of sets
In mathematics a field of sets is a pair \langle X, \mathcal \rangle where X is a set and \mathcal is an algebra over X i.e., a non-empty subset of the power set of X closed under the intersection and union of pairs of sets and under complements of individual sets.
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Finite ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.
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Galois theory
In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.
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Graph (discrete mathematics)
In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Group homomorphism
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
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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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Identity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group.
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Identity element
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.
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Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL(2, '''C''').
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List of small groups
The following list in mathematics contains the finite groups of small order up to group isomorphism.
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Lodovico Ferrari
Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian mathematician.
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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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Milton Babbitt
Milton Byron Babbitt (May 10, 1916 – January 29, 2011) was an American composer, music theorist, and teacher.
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Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).
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Multiplicative group of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to n from the set \ of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which in this ring are exactly those coprime to n. This group, usually denoted (\mathbb/n\mathbb)^\times, is fundamental in number theory.
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Musical composition
Musical composition can refer to an original piece of music, either a song or an instrumental music piece, the structure of a musical piece, or the process of creating or writing a new song or piece of music.
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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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Order (group theory)
In group theory, a branch of mathematics, the term order is used in two unrelated senses.
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Oxford University Press
Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.
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Positive real numbers
In mathematics, the set of positive real numbers, \mathbb_.
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Power set
In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.
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Presentation of a group
In mathematics, one method of defining a group is by a presentation.
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Quartic function
In algebra, a quartic function is a function of the form where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial.
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Quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication.
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure.
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Radical of an algebraic group
The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.
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Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.
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Representation theory of the symmetric group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained.
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Resolvent (Galois theory)
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root.
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Rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a simple (non-self-intersecting) quadrilateral whose four sides all have the same length.
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Split-complex number
In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z.
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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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Square
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted.
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Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
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Symmetric difference
In mathematics, the symmetric difference, also known as the disjunctive union, of two sets is the set of elements which are in either of the sets and not in their intersection.
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Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
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Symmetry group
In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.
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Twelve-tone technique
Twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition devised by Austrian composer Arnold Schoenberg (1874–1951) and associated with the "Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence.
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Unit (ring theory)
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.
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Dihedral group of order 4, Finite Group C2xC2, Four-group, Klein 4 group, Klein 4-Group, Klein 4-group, Klein four group, Klein group, Klein-4 group, Klein-four group, Klien group, Vierergruppe, Viergruppe.
References
[1] https://en.wikipedia.org/wiki/Klein_four-group
