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.
3-Group Theory-Mathematical Groups
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Date Modified: 2022.0225