Editing Maths (section)

==='''Common Groups'''===
* '''Aₙ - Alternating Group''' - The group of even permutations of a finite set. A₄ is symmetries of a Tetrahedron. - "Fearless Symmetry"
* '''Cₙ - Cyclic Group''' - 
* '''Dₙ - Dihedral Groups''' - A '''dihedral group''' is the group of '''symmetries''' of a regular polygon, which includes '''rotations and reflections'''. '''D₃''' is the '''dihedral group''' of the '''triangle'''. '''D₁≅Z₂''', '''D₂≅K₄'''.
* '''E⁺(n)''' -  The '''direct isometries''', i.e., isometries '''preserving orientation''', also called '''rigid motions'''; they are the moves of a rigid body in n-dimensional space. These include the translations, and the rotations, which together generate '''E⁺(n)'''. Also called a '''special Euclidean group''', and denoted '''SE(n)'''.
* '''T - Translation Group''' - T is a normal subgroup of E(n). T ◅ E(n).
* '''Sₙ - Symmetry Groups''' - '''S₃''' the '''symmetric group''' of 3 elements. '''S₃''' is '''isomorphic''' to '''D₃'''. ('''S₃ ≅ D₃''')
* '''O(n) - Orthogonal group''' - Reflections & rotations. The group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations ('''∘'''). It is equivalent to the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose. O(n, ℝ) is a subgroup of E(n), contains those that leave the origin fixed.
* '''SO(n) - Special Orthogonal''' - Rotation group. '''SO(2)''' is the [https://en.wikipedia.org/wiki/Circle_group circle group]. '''SO(3)''' is the group of all rotations about the origin of three-dimensional Euclidean space '''ℝ³''' under the operation of composition ('''∘'''). eg '''Roll, Pitch, Yaw'''.   '''Non-commutative'''. '''SO(n) ≤ O(n)'''.
* '''POₙ(ℝ)''' - '''Projective orthogonal group'''. '''PSO(V)''' for '''projective special orthogonal group'''.
* '''GLₙ(𝔽) - General Linear''' - '''Linear''' means '''matrix'''. The group of all n×n matrices with non-zero determinants under matrix multiplication. '''nonabelian''' group because matrix operations are non-commutative. The set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. The group '''GLₙ(ℝ)''' over the field of real numbers is a real '''Lie group''' of dimension '''n²''' as the set of all n×n real matrices, '''Mₙ(ℝ)''', forms a real vector space of dimension '''n²'''.
* '''Special Linear: SL(n)''' - Is the subgroup of '''GL(n, 𝔽)''' consisting of '''matrices with a determinant of 1'''. These form a group because the product of two matrices with determinant 1 again has determinant 1.
* '''ΓLₙ(ℝ)''' - General semilinear group. Contains '''GL'''.
* '''Sp(n)''' - The '''compact symplectic''' group '''Sp(n)''' is often written as '''USp(2n)''', indicating the fact that it is '''isomorphic''' to the '''group of unitary symplectic matrices'''.
* '''Sp(2n, 𝔽)''' - The '''symplectic group''' of degree '''2n''' over a field '''𝔽''' is the group of '''2n × 2n symplectic matrices''' with entries in '''𝔽''', and with the group operation that of matrix multiplication. Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group '''SL(2n, 𝔽)'''.
* '''Uₙ(ℝ) - The unitary group''' - A group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group '''SUₙ(ℝ)'''.
* <math>\Sigma_A</math> - The permutation group of set A.
* ℚᵃˡᵍ - The solutions of polynomials. Is a '''field'''. Its Elements are called "'''algebraic numbers'''". It is a subset of ℂ. - Fearless Symmetry p48


* [https://en.wikipedia.org/wiki/Classical_group Wikipedia: Classical group]
* [https://en.wikipedia.org/wiki/List_of_finite_simple_groups Wikipedia: List of finite simple groups]
* [https://en.wikipedia.org/wiki/List_of_simple_Lie_groups Wikipedia: List of simple Lie groups]