Group is Symmetry

Landau's book "Symmetry" explains it as follow:

Automorphism = Congruence= 叠合 has
1). Proper 真叠合 (symmetry: left= left, right = right)
2). Improper 非真叠合 (non-symmetry: reflection: left changed to right, vice-versa).
Congruence => preserve size / length
=> Movement 运动 (translation 平移, rotation about O )
= Proper congruence (Symmetry)

In Space S, the Automorphism that preserves the structure of S forms a Group Aut(G).
=> Group Aut(G) describes the Symmetry of Space S.

Hence Group is the language to describe Symmetry.

 

Automorphic Number

Automorphic Number (n)

Automorphism φ
φ: n -> nxn
nxn = {...}n

Example: 1, 5, 6, 25 are Automorphic Numbers
5x5 =25 ={2}5
6x6 =36 ={3}6
25x25 =625 ={6}25

Automorphism = Symmetry

Automorphism of a Set is an expression of its SYMMETRY.
1. Geometry figure (e.g. triangle) under certain transformations (reflection, rotation, ...), it is mapped upon itself, certain properties (distance, angle, relative location) are preserved.
=> the figure admits certain automorphism relative to its properties.
2. Automorphism of an arbitrary Set (with arbitrary relations between its elements) form an Automorphism Group of the set.