Symmetry in mathematics


Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations.
Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This can occur in many ways; for example, if X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points.
In general, every kind of structure in mathematics will have its own kind of symmetry, many of which are listed in the given points mentioned above.

Symmetry in geometry

The types of symmetry considered in basic geometry include reflectional symmetry, rotational symmetry, translational symmetry and glide reflection symmetry, which are described more fully in the main article Symmetry.

Symmetry in calculus

Even and odd functions

Even functions

Let f be a real-valued function of a real variable, then f is even if the following equation holds for all x and -x in the domain of f:
Geometrically speaking, the graph face of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis. Examples of even functions include, x2, x4, cos, and cosh.

Odd functions

Again, let f be a real-valued function of a real variable, then f is odd if the following equation holds for all x and -x in the domain of f:
That is,
Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin. Examples of odd functions are x, x3, sin, sinh, and erf.

Integrating

The integral of an odd function from −A to +A is zero, provided that A is finite and that the function is integrable.
The integral of an even function from −A to +A is twice the integral from 0 to +A, provided that A is finite and the function is integrable. This also holds true when A is infinite, but only if the integral converges.

Series

Symmetry in matrices

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if
By the definition of matrix equality, which requires that the entries in all corresponding positions be equal, equal matrices must have the same dimensions. Consequently, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if the entries are written as A =, then aij = aji, for all indices i and j.
For example, the following 3×3 matrix is symmetric:
Every square diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.

Symmetry in abstract algebra

Symmetric groups

The symmetric group Sn is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself. Since there are n! possible permutations of a set of n symbols, it follows that the order of the symmetric group Sn is n!.

Symmetric polynomials

A symmetric polynomial is a polynomial P in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2,..., n, one has P, Xσ,..., Xσ = P.
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view, the elementary symmetric polynomials are the most fundamental symmetric polynomials. A theorem states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that every symmetric polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.

Examples

In two variables X1 and X2, one has symmetric polynomials such as:
and in three variables X1, X2 and X3, one has as a symmetric polynomial:

  • Symmetric tensors

In mathematics, a symmetric tensor is tensor that is invariant under a permutation of its vector arguments:
for every permutation σ of the symbols.
Alternatively, an rth order symmetric tensor represented in coordinates as a quantity with r indices satisfies
The space of symmetric tensors of rank r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics.

Galois theory

Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it may be that for two of the roots, say A and B, that. The central idea of Galois theory is to consider those permutations of the roots having the property that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. An important proviso is that we restrict ourselves to algebraic equations whose coefficients are rational numbers. Thus, Galois theory studies the symmetries inherent in algebraic equations.

Automorphisms of algebraic objects

In abstract algebra, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

Examples

  • In set theory, an arbitrary permutation of the elements of a set X is an automorphism. The automorphism group of X is also called the symmetric group on X.
  • In elementary arithmetic, the set of integers, Z, considered as a group under addition, has a unique nontrivial automorphism: negation. Considered as a ring, however, it has only the trivial automorphism. Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.
  • A group automorphism is a group isomorphism from a group to itself. Informally, it is a permutation of the group elements such that the structure remains unchanged. For every group G there is a natural group homomorphism G → Aut whose image is the group Inn of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.
  • In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL.
  • A field automorphism is a bijective ring homomorphism from a field to itself. In the cases of the rational numbers and the real numbers there are no nontrivial field automorphisms. Some subfields of R have nontrivial field automorphisms, which however do not extend to all of R. In the case of the complex numbers, C, there is a unique nontrivial automorphism that sends R into R: complex conjugation, but there are infinitely many "wild" automorphisms. Field automorphisms are important to the theory of field extensions, in particular Galois extensions. In the case of a Galois extension L/''K the subgroup of all automorphisms of L'' fixing K pointwise is called the Galois group of the extension.