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