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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. Common examples of spaces that groups act on are sets, vector spaces, and topological spaces. Actions of groups on vector spaces are called representations of the group. When there is a natural correspondence between the set of group elements and the set of space transformations, a group can be interpreted as acting on the space in a canonical way. For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another (not necessarily distinct) element of the set
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Locally Compact Space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.Contents1 Formal definition 2 Examples and counterexamples2.1 Compact Hausdorff spaces 2.2 Locally compact Hausdorff spaces that are not compact 2.3 Hausdorff spaces that are not locally compact 2.4 Non-Hausdorff examples3 Properties3.1 The point at infinity 3.2 Locally compact groups4 Notes 5 ReferencesFormal definition[edit] Let X be a topological space. Most commonly X is called locally compact, if every point of X has a compact neighbourhood. There are other common definitions: They are all equivalent if X is a Hausdorff space
Hausdorff space
(or preregular). But they are not equivalent in general:1. every point of X has a compact neighbourhood. 2. every point of X has a closed compact neighbourhood. 2′
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Orthogonal Group
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose. An important subgroup of O(n) is the special orthogonal group, denoted SO(n), of the orthogonal matrices of determinant 1. This group is also called the rotation group, because, in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3)
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Identity Transformation
In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f(x) = x.Contents1 Definition 2 Algebraic property 3 Properties 4 See also 5 ReferencesDefinition[edit] Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfiesf(x) = x   for all elements x in M.[1]In other words, the function value f(x) in M (that is, the codomain) is always the same input element x of M (now considered as the domain)
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Equilateral Triangle
In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. They are regular polygons, and can therefore also be referred to as regular triangles.Contents1 Principal properties1.1 Equal cevians 1.2 Coincident triangle centers 1.3 Six triangles formed by partitioning by the medians 1.4 Points in the plane2 Notable theorems 3 Other properties 4 Geometric construction 5 Derivation of area formula5.1 Using the Pythagorean theorem 5.2 Using trigonometry6 In culture and society 7 See also 8 References 9 External linksPrincipal properties[edit]An equilateral triangle
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Abstraction (mathematics)
Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3] Two of the most highly abstract areas of modern mathematics are category theory and model theory.Contents1 Description 2 See also 3 References 4 Further readingDescription[edit] Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures
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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. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.[2] Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises
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Bijective Function
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements
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Inverse Function
In mathematics, an inverse function (or anti-function[1]) is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa. I.e., f(x) = y if and only if g(y) = x.[2][3] As a simple example, consider the real-valued function of a real variable given by f(x) = 5x − 7. Thinking of this as a step-by-step procedure (namely, take a number x, multiply it by 5, then subtract 7 from the result), to reverse this and get x back from some output value, say y, we should undo each step in reverse order. In this case that means that we should add 7 to y and then divide the result by 5. In functional notation this inverse function would be given by, g ( y ) = y + 7 5 . displaystyle g(y)= frac y+7 5
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Special Linear Group
In mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant det : GL ⁡ ( n , F ) → F × . displaystyle det colon operatorname GL (n,F)to F^ times
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Unit Sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point. Usually a specific point has been distinguished as the origin of the space under study and it is understood that a unit sphere or unit ball is centered at that point. Therefore one speaks of "the" unit ball or "the" unit sphere. For example, a one-dimensional sphere is the surface of what is commonly called a "circle", while such a circle's interior and surface together are the two-dimensional ball. Similarly, a two-dimensional sphere is the surface of the Euclidean solid known colloquially as a "sphere", while the interior and surface together are the three-dimensional ball. A unit sphere is simply a sphere of radius one
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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 h ( u ∗ v ) = h ( u ) ⋅ h ( v ) displaystyle h(u*v)=h(u)cdot h(v) 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 h ( u − 1 ) = h ( u ) − 1 . displaystyle hleft(u^ -1 right)=h(u)^ -1 ., Hence one
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If And Only If
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false). It is controversial whether the connective thus defined is properly rendered by the English "if and only if", with its pre-existing meaning
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Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.[1] An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix. The definition of kernel takes various forms in various contexts. But in all of them, the kernel of a homomorphism is trivial (in a sense relevant to that context) if and only if the homomorphism is injective
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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. In other words, a subgroup H of a group G is normal in G if and only if gH = Hg for all g in G. The definition of normal subgroup implies that the sets of left and right cosets coincide. In fact, a seemingly weaker condition that the sets of left and right cosets coincide also implies that the subgroup H of a group G is normal in G[1]
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Factor 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. For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written G / N, where G is the original group and N is the normal subgroup. (This is pronounced "G mod N", where "mod" is short for modulo.) Much of the importance of quotient groups is derived from their relation to homomorphisms
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