“group”的概念、定义、翻译、参考文献-科学参考

group

Mathematics

An operation on a set is worth considering only if it has properties likely to lead to interesting and useful results. Certain basic properties recur in different parts of mathematics and, if these are recognized, use can be made of the similarities that exist in the different situations. One such set of basic properties is specified in the definition of a group. The following, then, are all examples of groups: the set of real numbers with addition, the set of non‐zero real numbers with multiplication, the set of 2×2 real matrices with matrix addition, the set of vectors in 3‐dimensional space with vector addition, the set of all bijections from a set S onto itself with composition, the four numbers 1, i, −1, −i with multiplication. The definition is as follows: a group is a set G closed under an operation ○ such that
1. (i) for all a, b and c in G, a ○ (b ○ c) = (a ○ b) ○ c,
2. (ii) there is an identity element e in G such that a ○ e = e ○ a = a for all a in G,
3. (iii) for each a in G, there is an inverse element a′ in G such that a ○ a′ = a′ ○ a = e.
The group may be denoted by (G, ○), when it is necessary to specify the operation, but it may be called simply the group G when the intended operation is clear.

Chemistry

1. See periodic table.
2. A mathematical structure consisting of a set of elements A, B, C, etc., for which there exists a law of composition, referred to as ‘multiplication’. Any two elements can be combined to give a ‘product’ AB.
1. (1) Every product of two elements is an element of the set.
2. (2) The operation is associative, i.e. A(BC) = (AB)C.
3. (3) The set has an element I, called the identity element, such that IA = AI = A for all A in the set.
4. (4) Each element of the set has an inverse A⁻¹ belonging to the set such that AA⁻¹ = A⁻¹A = I.
Although the law of combination is called ‘multiplication’ this does not necessarily have its usual meaning. For example, the set of integers forms a group if the law of composition is addition.
Two elements A, B of a group commute if AB = BA. If all the elements of a group commute with each other the group is said to be Abelian. If this is not the case the group is said to be non-Abelian.
The interest of group theory in physics and chemistry is in analysing symmetry. Discrete groups have a finite number of elements, such as the symmetries involved in rotations and reflections of molecules, which give rise to point groups. Continuous groups have an infinite number of elements where the elements are continuous. An example of a continuous group is the set of rotations about a fixed axis. The rotation group thus formed underlies the quantum theory of angular momentum, which has many applications to atoms and nuclei.

Computer

A set G on which there is defined a dyadic operation ° (mapping G × G into G) that satisfies the following properties:
1. (a) ° is associative;
2. (b) ° has an identity, i.e. there is a unique element e in G with the property that
  $x ° e = e ° x = x$
  for all x in G; e is called the identity of the group;
3. (c) inverses exist in G, i.e. for each x in G there is an inverse, denoted by x⁻¹, with the property that
  $x ° x^{- 1} = x^{- 1} ° x = e$
  These are the group axioms.
Certain kinds of groups are of particular interest. If the dyadic operation ° is commutative, the group is said to be a commutative group or an abelian group (named for the Norwegian mathematician Niels Abel).
If there is only a finite number of elements n in the group, the group is said to be finite; n is then the order of the group. Finite groups can be represented or depicted by means of a Cayley table.
If the group has a generator then it is said to be cyclic; a cyclic group must be abelian.
The group is a very important algebraic structure that underlies many other algebraic structures such as rings and fields. There are direct applications of groups in the study of symmetry, in the study of transformations and in particular permutations, and also in error detecting and error correcting as well as in the design of fast adders.
Groups were originally introduced for solving an algebraic problem. By group theory it can be shown that algorithmic methods of a particular kind cannot exist for finding the roots of a general polynomial of degree greater than four. See also semigroup.

Geology and Earth Sciences

1. A number of geophones whose output is summed to feed one seismic channel. A particularly large number of geophones used per channel may be referred to as a ‘patch’. See also array.
2. See formation.

单词	group
释义	group Mathematics An operation on a set is worth considering only if it has properties likely to lead to interesting and useful results. Certain basic properties recur in different parts of mathematics and, if these are recognized, use can be made of the similarities that exist in the different situations. One such set of basic properties is specified in the definition of a group. The following, then, are all examples of groups: the set of real numbers with addition, the set of non‐zero real numbers with multiplication, the set of 2×2 real matrices with matrix addition, the set of vectors in 3‐dimensional space with vector addition, the set of all bijections from a set S onto itself with composition, the four numbers 1, i, −1, −i with multiplication. The definition is as follows: a group is a set G closed under an operation ○ such that (i) for all a, b and c in G, a ○ (b ○ c) = (a ○ b) ○ c, (ii) there is an identity element e in G such that a ○ e = e ○ a = a for all a in G, (iii) for each a in G, there is an inverse element a′ in G such that a ○ a′ = a′ ○ a = e. The group may be denoted by (G, ○), when it is necessary to specify the operation, but it may be called simply the group G when the intended operation is clear. Chemistry 1. See periodic table. 2. A mathematical structure consisting of a set of elements A, B, C, etc., for which there exists a law of composition, referred to as ‘multiplication’. Any two elements can be combined to give a ‘product’ AB. (1) Every product of two elements is an element of the set. (2) The operation is associative, i.e. A(BC) = (AB)C. (3) The set has an element I, called the identity element, such that IA = AI = A for all A in the set. (4) Each element of the set has an inverse A⁻¹ belonging to the set such that AA⁻¹ = A⁻¹A = I. Although the law of combination is called ‘multiplication’ this does not necessarily have its usual meaning. For example, the set of integers forms a group if the law of composition is addition. Two elements A, B of a group commute if AB = BA. If all the elements of a group commute with each other the group is said to be Abelian. If this is not the case the group is said to be non-Abelian. The interest of group theory in physics and chemistry is in analysing symmetry. Discrete groups have a finite number of elements, such as the symmetries involved in rotations and reflections of molecules, which give rise to point groups. Continuous groups have an infinite number of elements where the elements are continuous. An example of a continuous group is the set of rotations about a fixed axis. The rotation group thus formed underlies the quantum theory of angular momentum, which has many applications to atoms and nuclei. Computer A set G on which there is defined a dyadic operation ° (mapping G × G into G) that satisfies the following properties: (a) ° is associative; (b) ° has an identity, i.e. there is a unique element e in G with the property that $x ° e = e ° x = x$ for all x in G; e is called the identity of the group; (c) inverses exist in G, i.e. for each x in G there is an inverse, denoted by x⁻¹, with the property that $x ° x^{- 1} = x^{- 1} ° x = e$ These are the group axioms. Certain kinds of groups are of particular interest. If the dyadic operation ° is commutative, the group is said to be a commutative group or an abelian group (named for the Norwegian mathematician Niels Abel). If there is only a finite number of elements n in the group, the group is said to be finite; n is then the order of the group. Finite groups can be represented or depicted by means of a Cayley table. If the group has a generator then it is said to be cyclic; a cyclic group must be abelian. The group is a very important algebraic structure that underlies many other algebraic structures such as rings and fields. There are direct applications of groups in the study of symmetry, in the study of transformations and in particular permutations, and also in error detecting and error correcting as well as in the design of fast adders. Groups were originally introduced for solving an algebraic problem. By group theory it can be shown that algorithmic methods of a particular kind cannot exist for finding the roots of a general polynomial of degree greater than four. See also semigroup. Geology and Earth Sciences 1. A number of geophones whose output is summed to feed one seismic channel. A particularly large number of geophones used per channel may be referred to as a ‘patch’. See also array. 2. See formation.
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