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.
Mathematical Sciences>Group Theory>Mathematical Groups>Subgroup
3-Group Theory-Mathematical Groups-Subgroup
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Date Modified: 2022.0224