Basics of Groups
A group is a mathematical object that consists of a set G together with a binary operation (denoted by *) that combines any two elements of G to form a third element in G. To be a group, the following axioms must be satisfied:
• Closure: For any a, b in G, the result of a * b is also in G.
• Associativity: For any a, b, c in G, (a * b) * c = a * (b * c).
• Identity element: There exists an element e in G, such that for any a in G, a * e = e * a = a.
• Inverse element: For every element a in G, there exists an element b in G (denoted by a^-1), such that a * b = b * a = e.
In other words, a group is a set equipped with an operation that satisfies certain properties. The operation must be associative, and there must exist an identity element and an inverse element for every element in the set.
Groups arise naturally in many areas of mathematics, including algebra, geometry, and number theory. They are used to study symmetry and transformations of objects, as well as to classify and understand mathematical structures. Group theory provides a powerful language and framework for describing and analyzing many mathematical systems and phenomena.
Some examples of groups include:
• The integers under addition.
• The real numbers excluding zero, under multiplication.
• The symmetric group on n elements, which consists of all permutations of n objects.
• The group of rotations and reflections of a regular n-gon.
Group theory has many important theorems and concepts, including group homomorphisms, subgroups, cosets, and normal subgroups. It also has connections to other areas of mathematics, including representation theory, algebraic geometry, and topology.
Definition of a group
A group is a mathematical object that consists of a set G together with a binary operation (denoted by *) that combines any two elements of G to form a third element in G. To be a group, the following axioms must be satisfied:
• Closure: For any a, b in G, the result of a * b is also in G.
• Associativity: For any a, b, c in G, (a * b) * c = a * (b * c).
• Identity element: There exists an element e in G, such that for any a in G, a * e = e * a = a.
• Inverse element: For every element a in G, there exists an element b in G (denoted by a^-1), such that a * b = b * a = e.
In other words, a group is a set equipped with an operation that satisfies certain properties. The operation must be associative, and there must exist an identity element and an inverse element for every element in the set.
Groups arise naturally in many areas of mathematics, including algebra, geometry, and number theory. They are used to study symmetry and transformations of objects, as well as to classify and understand mathematical structures. Group theory provides a powerful language and framework for describing and analyzing many mathematical systems and phenomena.
Some examples of groups include:
• The integers under addition.
• The real numbers excluding zero, under multiplication.
• The symmetric group on n elements, which consists of all permutations of n objects.
• The group of rotations and reflections of a regular n-gon.
Group theory has many important theorems and concepts, including group homomorphisms, subgroups, cosets, and normal subgroups. It also has connections to other areas of mathematics, including representation theory, algebraic geometry, and topology.
Examples of groups (e.g. permutation groups, matrix groups)
There are many examples of groups in mathematics. Here are a few:
• Permutation groups: A permutation group is a group that consists of all permutations of a set. For example, the symmetric group on n elements, denoted by Sn, is the group of all permutations of {1, 2, ..., n}. This group has order n!, which is the number of ways to order n distinct objects.
• Matrix groups: A matrix group is a group that consists of all invertible matrices of a certain size over a given field. For example, the special orthogonal group SO(n) is the group of all n×n orthogonal matrices with determinant 1. This group represents rotations in n-dimensional space.
• Abelian groups: An Abelian group is a group in which the binary operation is commutative. For example, the integers under addition form an Abelian group.
• Finite groups: A finite group is a group with a finite number of elements. There are many examples of finite groups, such as the dihedral groups, the cyclic groups, and the alternating groups.
• Lie groups: A Lie group is a group that is also a smooth manifold, meaning it has a well-behaved notion of differentiation. Examples include the special linear group SL(n), the general linear group GL(n), and the unitary group U(n).
These are just a few examples of the many types of groups that arise in mathematics. Groups are used to study symmetry and transformation properties of objects, as well as to classify and understand mathematical structures. Group theory provides a powerful language and framework for describing and analyzing many mathematical systems and phenomena.
Group properties (identity element, inverse elements, associativity)
Groups have several important properties that must be satisfied for a set G to be considered a group under an operation *:
• Identity element: A group must have an identity element e, which is an element of G such that for any a in G, a * e = e * a = a. In other words, the identity element leaves other elements unchanged when combined with them. The identity element is unique within the group, meaning that there can be no other element that satisfies the definition of an identity element.
• Inverse elements: For every element a in G, there must exist an element b in G, denoted by a^-1, such that a * b = b * a = e. In other words, every element in G must have an inverse element that undoes the effect of combining it with another element. The inverse element is unique within the group, meaning that there can be no other element that satisfies the definition of an inverse element.
• Associativity: The operation * in a group must be associative, meaning that for any a, b, and c in G, (a * b) * c = a * (b * c). In other words, the order in which operations are performed does not matter.
These properties ensure that the binary operation * behaves in a well-defined way, allowing for meaningful computations and analysis of the group structure. They also ensure that certain fundamental results hold, such as the cancellation law and the fact that the product of any finite number of group elements is well-defined.
Cayley tables and group presentations
Two common ways of describing a group are through Cayley tables and group presentations.
A Cayley table is a way of representing the binary operation of a group by listing all the possible products of elements in the group. For a group with n elements, the table is an n x n square, where the rows and columns are labeled with the group elements and the entries in the table give the product of the corresponding row and column elements. The identity element of the group is usually listed in the upper-left corner of the table, and each row and column should contain each element of the group exactly once.
For example, consider the group Z4 under addition modulo 4. The Cayley table for this group is:
diffCopy code
+ | 0 1 2 3
--+--------
0 | 0 1 2 3
1 | 1 2 3 0
2 | 2 3 0 1
3 | 3 0 1 2
This table shows that the group operation is commutative (since the table is symmetric across the main diagonal), that the identity element is 0 (since every element combined with 0 gives itself), and that every element has an inverse (since each row and column contains each element exactly once).
A group presentation is a way of describing a group using generators and relations. Generators are a set of elements that can be used to generate all the elements of the group by combining them using the group operation. Relations are equations that describe how certain combinations of generators are related to each other. A group presentation is denoted by G = <S|R>, where S is the set of generators and R is the set of relations.
For example, consider the group Z under addition. This group has a single generator 1, and the only relation is that 1 + (-1) = 0. Therefore, the group presentation for Z is G = <1 | 1 + (-1) = 0>.
Group presentations can be useful for understanding the underlying structure of a group, and for comparing different groups with similar presentations. They can also be used to study groups that are too large or too complicated to describe explicitly using Cayley tables.
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