Element sets {group, mathematics} {mathematical group}, such as equilateral-triangle points, can have one or more operations, such as rotations through 60-degree angles. All elements are products {word, group} of the operation {generator, group} acting on group elements.
elements
Elements can be geometric or algebraic. Groups have finite or infinite number {order, group} of elements.
operation
Operation maps element to element {product, operation}. For example, rotation maps point to second point. Addition maps two elements to one element {binary operation}.
generator
Some groups have constraints on generators, and some groups (free group) have no constraints on generators.
identity
Operations can map element to same element {identity operation}.
inverse
Operations can be another operation's opposite {inverse operation}. Operation followed by inverse operation makes original element.
association
For three elements and a binary operation, do operation on first and second elements, and then do operation on product element and third element, (a + b) + c, or do operation on second and third elements, and then do operation on first element and product element, a + (b + c). The resulting products can be the same (associative group) {association, group} or different {non-associative group}.
commutation
Binary operations can happen in either order: a + b or b + a. For two elements and a binary operation, do operation on first and second elements, a + b, or do operation on second and first elements, b + a. The resulting products can be the same (commutative group) {commutation, group} or different {non-commutative group}. Non-commutative operations typically make the negative: a + b = -(b + a). Complex-number multiplication is commutative. Matrix multiplication is non-commutative.
transformation groups
Different spaces and geometries have different invariant transformations. For example, different spaces and geometries can be invariant under rigid motions {motion geometry}. For different space geometries, number of dimensions determines number of possible shape changes. Space geometries are in a hierarchy from least general to most general: metrical geometry, affine geometry, projective geometry, and topology.
Transformation groups have operations that are linear transformations {analytic transformation, group}. Different groups have different invariant transformations. Transformation groups describe spaces and geometries. For example, group U(1) describes circle rotation symmetries (electromagnetism has U(1) symmetry, which implies charge conservation). Symmetry groups {universality class} represent substances with different symmetries.
Transformation results can be in tables {multiplication table}. Rows and columns are elements, and cells have products.
types
Transformation groups are finite continuous groups. Differential equations make an infinite continuous group. Substitution groups are finite-order discontinuous groups. Automorphic functions are infinite-order discontinuous groups.
matrix
Matrices can represent groups.
Mathematical Sciences>Group Theory
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Date Modified: 2022.0224