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Normal Subgroup
In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ..., a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup In group theory, a branch of mathematics, given a group (mathematics), 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 ∗. More precisely ... that is invariant under conjugation Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ... by members of the group A group is a number A number is a mathematical objec ...
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Commutator Subgroup
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., more specifically in abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ..., the commutator subgroup or derived subgroup of a group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ... is the subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics ...
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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 of groups, direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in which a Group (mathematics), group can be made up of two subgroups, one of which is a normal subgroup. * an ''outer'' semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as ''semidirect products''. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as Group_extension#Classifying_split_extensions , splitting extension). Inner semidirect product definitions Given a ...
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Direct Product Of Groups
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., specifically in group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ..., the direct product is an operation that takes two groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ... and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product In ...
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T-group (mathematics)
In mathematics, in the field of group theory, a T-group is a group (mathematics), group in which the property of normal subgroup, normality is transitive, that is, every subnormal subgroup is normal. Here are some facts about T-groups: *Every simple group is a T-group. *Every quasisimple group is a T-group. *Every abelian group is a T-group. *Every Hamiltonian group is a T-group. *Every nilpotent group, nilpotent T-group is either abelian or Hamiltonian, because in a nilpotent group, every subgroup is subnormal. *Every normal subgroup of a T-group is a T-group. *Every homomorphic image of a T-group is a T-group. *Every solvable group, solvable T-group is metabelian group, metabelian. The solvable T-groups were characterized by Wolfgang Gaschütz as being exactly the solvable groups ''G'' with an abelian normal Hall subgroup ''H'' of odd group order, order such that the quotient group ''G''/''H'' is a Dedekind group and ''H'' is Group action (mathematics), acted upon by Conjugacy clas ...
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Dihedral Group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the n-gon, -gon, a group of order . In abstract algebra, refers to this same dihedral group. The geometric convention is used in this article. Definition Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetry, rotational symmetries and n reflection symmetry, reflection symmetries. Usually, we take n \ge 3 here. The associated rotations and reflection (mathematics), reflections make up the dihedral group \mathrm_n. If n is odd, each axis of symmetry connects the midpoint of one side to t ...
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Transitive Relation
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion Inclusion or Include may refer to: Sociology * Social inclusion, affirmative action to change the circumstances and habits that leads to s ... as well as each equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... needs to be transitive. Definition ...
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Rotation
Rotation is the circular movement of an object around an ''axis of rotation Rotation around a fixed axis is a special case of rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotati ...''. A three-dimensional object may have an infinite number of rotation axes. If the rotation axis passes internally through the body's own center of mass In physics, the center of mass of a distribution of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ..., then the body is said to be ''autorotating'' or ''spinning Spin or spinning may refer to: Businesses * SPIN (cable system) SPIN (or South Pacific Island Network) was a submarine communications cable, submarine communications cable system that would ...
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Euclidean Group
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a Euclidean group is the group of (Euclidean) isometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... of a Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ... \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance In mathematics Mathematics (from Ancient ...
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Translation Group
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same Distance geometry, distance in a given direction. A translation can also be interpreted as the addition of a constant vector space, vector to every point, or as shifting the Origin (mathematics), origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image (mathematics), image of a subset A under the function (mathematics), function T is the translate of A by T . The translate of A by T_ is often written A+\mathbf . Horizontal and vertical translations In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction ...
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Rubik's Cube Group
The Rubik's Cube group is a Group (mathematics), group (G, \cdot ) that represents the Mathematical structure, structure of the Rubik's Cube mechanical puzzle. Each element of the Set (mathematics), set G corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces. With this representation, not only can any cube move be represented, but also any position of the cube as well, by detailing the cube moves required to rotate the solved cube into that position. Indeed with the solved position as a starting point, there is a Bijection, one-to-one correspondence between each of the legal positions of the Rubik's Cube and the elements of G. The group Binary operation, operation \cdot is the Function composition, composition of cube moves, corresponding to the result of performing one cube move after another. The Rubik's Cube group is constructed by labeling each of the 48 non-center facets with the integers 1 to 48. Each configuration of the cube can ...
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Symmetric Group
In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ..., the symmetric group defined over any set is the group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ... whose elements are all the bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...s from the set to itself, and whose group operation form the Rubik's Cube group. In mathematics Mathematics ...
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