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.
Mathtuition88 