# Micmaths : Math Jargon “Automorphism”  $z = a + i.b$ $\bar {z} = a - i.b$ $\phi : z \to \bar {z}$ $\phi = Automorphism$ 自同构

[French]

– by Mickael Launay (École Normale Supérieure) , author of the best seller book 《Le Grand Roman des Maths》, 《万物皆数》translated into 16 languages.

# Morphism Summary Chart

The more common morphisms are:

1. Homomorphism (Similarity between 2 different structures) 同态
Analogy: Similar triangles of 2 different triangles.

2. Isomorphism (Sameness between 2 different structures) 同构
Analogy: Congruence of 2 different triangles

Example: 2 objects are identical up to an isomorphism.

3. Endomorphism (Similar structure of self) = {Self + Homomorphism} 自同态
Analogy: A triangle and its image in a magnifying glass.

4. Automorphism (Sameness structure of self) = {Self + Isomorphism} 自同构
Analogy: A triangle and its image in a mirror; or
A triangle and its rotated (clock-wise or anti-clock-wise), or reflected (flip-over) self. 5. Monomorphism 单同态 = Injective + Homomorphism 6. Epimorphism 满同态 = Surjective + Homomorphism

# 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.