Groups {Abelian group} {commutative group} can have commutative operation.
Groups {associative group} can have associative operation.
Groups {continuous group} can have local operations on infinitesimal elements, to make real-number algebras.
Continuous commutative or non-commutative groups {Lie group}, such as n^2 parameter Lie groups {SU(n) group}, have parameters and make associative and non-associative algebras (Lie algebra).
Groups {non-Abelian group} can have non-commutative operations.
Classical groups {symmetric bilinear form} {orthogonal group} can preserve linear vector transformations in three-dimensional Euclidean space, because axes stay at right angles and maintain non-singular quadratic form.
matrices
Matrices can represent orthogonal groups. For orthogonal-group transformations, inverse matrix equals transpose matrix.
Matrix diagonal has 0, 1, or -1. Orthogonal groups {Lorentz group} can have diagonal {Lorentzian diagonal} with one 1, zero 0s, and non-zero -1s. Number of 1s and -1s can be invariant {signature, group}.
If diagonal has only 1s {positive-definite diagonal}, matrix is non-singular and positive, and has positive-definite non-singular symmetric tensors {metric, tensor}. Group {pseudo-orthogonal group} matrix can be singular and not positive-definite {pseudometric}.
If orthogonal-group transformation determinant equals 1, group is non-reflective.
Groups {substitution group} {permutation group} can have operation that maps one element to another element {substitution, group}. An example is rearranging element positions {permutation, group}, such as exchanging elements at line-segment ends. Permutations can result in equivalent subsets {combination, group}. Product element is in group {closure, group} {closed group}.
identity
Substitution groups must have identity operation.
inverse
Substitution-group operations must have inverse operations.
transitive
Group operation, such as rotations, can substitute all elements {transitive group}. Group operations, such as division, may not apply to all elements or some elements may not be operation results {intransitive group}.
primitive
Elements can have separate subsets that permute among themselves {imprimitive group} or have no separable subsets {primitive group}.
abstract groups
Generalized substitution groups {abstract group} show group properties.
Groups {symmetry group, mathematics} can have operations that leave system unchanged {symmetry transformation}.
types
If elements are alphabet characters, groups {symmetric group} can have all letter permutations, or groups {alternating group} can have all even permutations. If elements are numbers, groups {cyclic group} can have all element powers. If elements are points, motions can go around angles {rotation symmetry}, can be axis linear motions {transformational symmetry}, or can be line motions {translation symmetry}. Transformations {displacement, transformation} can preserve geometric-figure congruence.
types: reflection
If elements are points, motions in planes can go through axes {reflection symmetry}. Reflections can be through points {radial symmetry reflection} {central symmetry reflection}, lines {axial symmetry reflection}, or planes {plane symmetry reflection}. If elements are points, motions can have both rotation and reflection {inversion symmetry}.
Classical groups {symplectic group} can have linear transformations with anti-symmetry (transpose equals negative of original).
Groups {topological group} can be continuous, infinite, connected, and compact.
Groups {transformation group} can be finite continuous groups.
Classical groups {unitary group} can preserve complex linear transformations and maintain non-singular Hermitean form, rather than orthogonal-group quadratic form. Hermitean form equals its complex conjugate {Hermitean conjugation}, which is the dual space. For unitary groups, linear-transformation inverse is positive-definite transpose. For pseudo-unitary groups, linear-transformation inverse is not positive-definite transpose.
Smaller groups {homomorphism, group} can have same elements and same multiplication table as larger-group subgroup. For example, C4v group is homomorphic to group of multiplications of 1 and -1. Several extended-group members can associate with same smaller-group member {many-to-one mapping}. Isomorphism and homomorphism are independent group properties.
Two groups can have same number of elements and same multiplication table {isomorphism, group}. For example, group of eight 2x2 matrices with elements 0 and 1 is isomorphic to C4v group of square symmetries. Isomorphism and homomorphism are independent group properties.
Groups can have smaller groups {subgroup}. If subgroups share only the identity element, and multiplication commutes, products of two subgroups are subgroups. Groups {composite group} can have invariant subgroups.
Group elements can be either non-subgroup elements or subgroup elements. All non-subgroup elements form a set {coset, group}.
Groups {factor group} can have normal subgroups that factor the group commutatively.
Groups {normal subgroup, commutation} {invariant subgroup} {self-conjugate subgroup} can have same products when using group member and then subgroup member, or subgroup member and then group member, so group elements commute with subgroup elements.
Subgroups {proper subgroup} can have more than one element and less than all elements.
Subgroup {quotient group} elements can be an invariant-subgroup's cosets.
Groups, such as complex numbers, can have subgroups. Groups, such as empty set and sets with one element, can have no subgroups. Non-Abelian Lie groups {simple group} can have no normal subgroups.
Non-Abelian Lie groups {semi-simple group} can have no Abelian normal subgroups.
Continuous simple groups {classical group} have four families and exceptional groups.
families
For m = 1, 2, 3, ... SU(m + 1) is unitary {Am family}, with dimension m * (m + 2). SO(2*m + 1) is orthogonal {Bm family}, with dimension m * (2*m + 1). Sp(m) is symplectic {Cm family}, with dimension m * (2*m + 1). SO(2*m) is orthogonal {Dm family}, with dimension m * (2*m - 1).
exceptional groups
Other continuous simple groups {exceptional groups} are E6, E7, E8, F4, and G2.
finite
Finite simple groups have classical groups and exceptional groups. SO(3) group is Special, because it has unit determinant and so is non-reflective. SO(3) group is Orthogonal, because the three axes are at right angles. SO(3) group has number 3 because rotations can be in three dimensions.
product
Simple groups can combine {product group} to make element pairs.
product: linear
n-dimensional vector spaces have product groups {general linear group} GL(n) of linear translational symmetries. GL(n) is product group of n x n singular matrices, one for each dimension.
Simple groups {sporadic group} can have finite number of elements, such as Janko J4, Fischer Fi24, Baby Monster B, Monster M, M12, M24, and Co1. There are 26 sporadic groups.
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Date Modified: 2022.0225