From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it maps inverses to inverses in the sense that h = h −1. Hence one can say that h is compatible with the group structure, older notations for the homomorphism h may be xh, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets and this approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right. In areas of mathematics where one considers groups endowed with additional structure, for example, a homomorphism of topological groups is often required to be continuous. The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure, an equivalent definition of group homomorphism is, The function h, G → H is a group homomorphism if whenever a ∗ b = c we have h ⋅ h = h.
In other words, the group H in some sense has an algebraic structure as G. Monomorphism A group homomorphism that is injective, i. e. preserves distinctness, epimorphism A group homomorphism that is surjective, i. e. reaches every point in the codomain. Isomorphism A group homomorphism that is bijective, i. e. injective and surjective and its inverse is a group homomorphism. In this case, the groups G and H are called isomorphic, endomorphism A homomorphism, h, G → G, the domain and codomain are the same. Also called an endomorphism of G. Automorphism An endomorphism that is bijective, the set of all automorphisms of a group G, with functional composition as operation, forms itself a group, the automorphism group of G. As an example, the group of contains only two elements, the identity transformation and multiplication with −1, it is isomorphic to Z/2Z. We define the kernel of h to be the set of elements in G which are mapped to the identity in H ker ≡. the kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism.
The first isomorphism theorem states that the image of a group homomorphism, if and only if ker =, the homomorphism, h, is a group monomorphism, i. e. h is injective. The map h, Z → Z/3Z with h = u mod 3 is a group homomorphism and it is surjective and its kernel consists of all integers which are divisible by 3. The exponential map yields a homomorphism from the group of real numbers R with addition to the group of non-zero real numbers R* with multiplication. The kernel is and the image consists of the real numbers. The exponential map yields a group homomorphism from the group of complex numbers C with addition to the group of non-zero complex numbers C* with multiplication. This map is surjective and has the kernel, as can be seen from Eulers formula, fields like R and C that have homomorphisms from their additive group to their multiplicative group are thus called exponential fields
In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication.
The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE
In mathematics, an operation is a calculation from zero or more input values to an output value. The number of operands is the arity of the operation, the most commonly studied operations are binary operations of arity 2, such as addition and multiplication, and unary operations of arity 1, such as additive inverse and multiplicative inverse. An operation of arity zero, or 0-ary operation is a constant, the mixed product is an example of an operation of arity 3, or ternary operation. Generally, the arity is supposed to be finite, but infinitary operations are sometimes considered, in this context, the usual operations, of finite arity are called finitary operations. There are two types of operations and binary. Unary operations involve only one value, such as negation and trigonometric functions, binary operations, on the other hand, take two values, and include addition, multiplication and exponentiation. Operations can involve mathematical objects other than numbers, the logical values true and false can be combined using logic operations, such as and, or, and not.
Vectors can be added and subtracted, rotations can be combined using the function composition operation, performing the first rotation and the second. Operations on sets include the binary operations union and intersection and the operation of complementation. Operations on functions include composition and convolution, operations may not be defined for every possible value. For example, in the numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined form a set called its domain, the set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its range. For example, in the numbers, the squaring operation only produces non-negative numbers. A vector can be multiplied by a scalar to form another vector, and the inner product operation on two vectors produces a scalar. An operation may or may not have certain properties, for example it may be associative, anticommutative, the values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output.
Operations can have fewer or more than two inputs, an operation is like an operator, but the point of view is different. An operation ω is a function of the form ω, V → Y, where V ⊂ X1 × … × Xk. The sets Xk are called the domains of the operation, the set Y is called the codomain of the operation, thus a unary operation has arity one, and a binary operation has arity two
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. This means that either object can be repositioned and reflected so as to coincide precisely with the other object, so two distinct plane figures on a piece of paper are congruent if we can cut them out and match them up completely. Turning the paper over is permitted, in elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects, two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure, two circles are congruent if they have the same diameter. The related concept of similarity applies if the objects differ in size, for two polygons to be congruent, they must have an equal number of sides. Two polygons with n sides are congruent if and only if they each have identical sequences side-angle-side-angle-. for n sides.
Congruence of polygons can be established graphically as follows, match, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that two vertices match. Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches, reflect the rotated figure about this matched side until the figures match. If at any time the step cannot be completed, the polygons are not congruent, two triangles are congruent if their corresponding sides are equal in length, in which case their corresponding angles are equal in measure. SSS, If three pairs of sides of two triangles are equal in length, the triangles are congruent, ASA, If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus, in most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems.
In the School Mathematics Study Group system SAS is taken as one of 22 postulates, AAS, If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, the triangles are congruent. For American usage, AAS is equivalent to an ASA condition, RHS, known as HL, If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, the triangles are congruent. The SSA condition which specifies two sides and a non-included angle does not by itself prove congruence, in order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. The opposite side is longer when the corresponding angles are acute. This is the case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, a familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7,00 now, 8 hours it will be 3,00. Usual addition would suggest that the time should be 7 +8 =15. Likewise, if the clock starts at 12,00 and 21 hours elapse, the time will be 9,00 the next day, because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below,12 is congruent not only to 12 itself, Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations on integers, addition and multiplication. For a positive n, two integers a and b are said to be congruent modulo n, written, a ≡ b.
The number n is called the modulus of the congruence, for example,38 ≡14 because 38 −14 =24, which is a multiple of 12. The same rule holds for negative values, −8 ≡72 ≡ −3 −3 ≡ −8. Equivalently, a ≡ b mod n can be thought of as asserting that the remainders of the division of both a and b by n are the same, for instance,38 ≡14 because both 38 and 14 have the same remainder 2 when divided by 12. It is the case that 38 −14 =24 is a multiple of 12. A remark on the notation, Because it is common to consider several congruence relations for different moduli at the same time, in spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been if the notation a ≡n b had been used. The properties that make this relation a congruence relation are the following, if a 1 ≡ b 1 and a 2 ≡ b 2, then, a 1 + a 2 ≡ b 1 + b 2 a 1 − a 2 ≡ b 1 − b 2. The above two properties would still hold if the theory were expanded to all real numbers, that is if a1, a2, b1, b2. The next property, would fail if these variables were not all integers, the notion of modular arithmetic is related to that of the remainder in Euclidean division.
The operation of finding the remainder is referred to as the modulo operation. For example, the remainder of the division of 14 by 12 is denoted by 14 mod 12, as this remainder is 2, we have 14 mod 12 =2
The material conditional is a logical connective that is often symbolized by a forward arrow →. The material conditional is used to form statements of the form
Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition and division. Multiplication can be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page.
In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard.
This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
In mathematics, when the elements of some set S have a notion of equivalence defined on them, one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the equivalence class if. Formally, given a set S and an equivalence relation ~ on S and it may be proven from the defining properties of equivalence relations that the equivalence classes form a partition of S. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of S by ~ and is denoted by S / ~. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories. If X is the set of all cars, and ~ is the relation has the same color as. X/~ could be identified with the set of all car colors. Let X be the set of all rectangles in a plane, for each positive real number A there will be an equivalence class of all the rectangles that have area A.
Consider the modulo 2 equivalence relation on the set Z of integers, x ~ y if and this relation gives rise to exactly two equivalence classes, one class consisting of all even numbers, and the other consisting of all odd numbers. Under this relation, and all represent the element of Z/~. Let X be the set of ordered pairs of integers with b not zero, the same construction can be generalized to the field of fractions of any integral domain. In this situation, each equivalence class determines a point at infinity, the equivalence class of an element a is denoted and is defined as the set = of elements that are related to a by ~. An alternative notation R can be used to denote the class of the element a. This is said to be the R-equivalence class of a, the set of all equivalence classes in X with respect to an equivalence relation R is denoted as X/R and called X modulo R. The surjective map x ↦ from X onto X/R, which each element to its equivalence class, is called the canonical surjection or the canonical projection map.
When an element is chosen in each class, this defines an injective map called a section. If this section is denoted by s, one has = c for every equivalence class c, the element s is called a representative of c. Any element of a class may be chosen as a representative of the class, there is a section that is more natural than the other ones
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. Although derived from a subgroup, cosets are not usually themselves subgroups of G, a coset is a left or right coset of some subgroup in G. Since Hg = g , the right coset Hg and the left coset g are the same, hence it is not meaningful to speak of a coset as being left or right unless one first specifies the underlying subgroup. In other words, a right coset of one subgroup equals a left coset of a different subgroup, if the left cosets and right cosets are the same, H is a normal subgroup and the cosets form a group called the quotient or factor group. The map gH ↦ −1 = Hg−1 defines a bijection between the left cosets and the cosets of H, so the number of left cosets is equal to the number of right cosets. The common value is called the index of H in G, for abelian groups, left cosets and right cosets are always the same. If the group operation is written additively, the notation used changes to g + H or H + g, Cosets are a basic tool in the study of groups, for example they play a central role in Lagranges theorem.
Let G = be the group formed by multiplication, which is isomorphic to C2. Then = H = H and = 1H = H1 are the cosets of H in G. Because its left and right cosets with respect to any element of G coincide, let G be the additive group of the integers, Z = and H the subgroup = where m is a positive integer. Then the cosets of H in G are the m sets mZ, mZ +1, there are no more than m cosets, because mZ + m = m = mZ. The coset is the class of a modulo m. Another example of a comes from the theory of vector spaces. The elements of a space form an abelian group under vector addition. It is not hard to show that subspaces of a space are subgroups of this group. For a vector space V, a subspace W, and a vector a in V, the sets are called affine subspaces. In terms of vectors, these affine subspaces are all the lines or planes parallel to the subspace. Some authors define the left cosets of H in G to be the equivalence classes under the relation on G given by x ~ y if
In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients and subobjects. Versions of the theorems exist for groups, vector spaces, Lie algebras, in universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences. Less general versions of these theorems can be found in work of Richard Dedekind, three years later, B. L. van der Waerden published his influential Algebra, the first abstract algebra textbook that took the groups-rings-fields approach to the subject. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly and we first state the three isomorphism theorems in the context of groups. Note that some sources switch the numbering of the second and third theorems, finally, in the most extensive numbering scheme, the lattice theorem is sometimes referred to as the fourth isomorphism theorem. Let G and H be groups, and let φ, G → H be a homomorphism.
Then, The kernel of φ is a subgroup of G, The image of φ is a subgroup of H. In particular, if φ is surjective H is isomorphic to G / ker, let S be a subgroup of G, and let N be a normal subgroup of G. Then the following hold, The product S N is a subgroup of G, The intersection S ∩ N is a subgroup of S. Technically, it is not necessary for N to be a normal subgroup, in this case, the intersection S ∩ N is not a normal subgroup of G, but it is still a normal subgroup of S. This isomorphism theorem has been called the diamond theorem due to the shape of the subgroup lattice with S N at the top, S ∩ N at the bottom. It has even called the parallelogram rule because in the resulting subgroup lattice the two sides assumed to represent the quotient groups / N and S / are equal in the sense of isomorphism. An example of an application of the second theorem is with projective linear groups. Setting G = G L2, S = S L2, let G be a group, and N a normal subgroup of G. Then If K is a subgroup of G such that N ⊆ K ⊆ G, every subgroup of G / N is of the form K / N, for some subgroup K of G such that N ⊆ K ⊆ G.
If K is a subgroup of G such that N ⊆ K ⊆ G. Every normal subgroup of G / N is of the form K / N, If K is a normal subgroup of G such that N ⊆ K ⊆ G, the quotient group / is isomorphic to G / K. W. R. Scott calls it the Freshman theorem because the result follows by cancellation of N
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, matrices, the conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert and Noether, rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry. A ring is a group with a second binary operation that is associative, is distributive over the abelian group operation. By extension from the integers, the group operation is called addition. Whether a ring is commutative or not has profound implications on its behavior as an abstract object, as a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory.
Its development has greatly influenced by problems and ideas occurring naturally in algebraic number theory. The most familiar example of a ring is the set of all integers, Z, −5, −4, −3, −2, −1,0,1,2,3,4,5. The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings, a ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms 1. R is a group under addition, meaning that, + c = a + for all a, b, c in R. a + b = b + a for all a, b in R. There is an element 0 in R such that a +0 = a for all a in R, for each a in R there exists −a in R such that a + =0. R is a monoid under multiplication, meaning that, · c = a · for all a, b, c in R. There is an element 1 in R such that a ·1 = a and 1 · a = a for all a in R.3. Multiplication is distributive with respect to addition, a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many follow a alternative convention in which a ring is not defined to have a multiplicative identity.
This article adopts the convention that, unless stated, a ring is assumed to have such an identity