Automorphism is an isomorphism from a group to itself.

2. What is/are Automorphism/s?
• An automorphism is an isomorphism from a group to
itself.
• The set of all automorphisms of G forms a group, called
the automorphism group of G, and denoted Aut(G).
An automorphism is simply a bijective homomorphism of an object with
itself.
An automorphism is an endomorphism (i.e., a morphism from an object to
itself) which is also an isomorphism (in the categorical sense of the word,
meaning there exists a right and left inverse endomorphism).

3. One of the earliest group
automorphisms (automorphism of a group,
not simply a group of automorphisms of
points) was given by the Irish
mathematician William Rowan Hamilton in
1856, in his icosian calculus, where he
discovered an order two
automorphism, writing:
so that 𝝁 is a new fifth root of unity, connected
with the former fifth root 𝝀 by relations of
perfect reciprocity.
Source: Wikipedia

4. • Homomorphism
preserve the group operation
𝜙 𝑥 ∙ 𝑦 = 𝜙 𝑥 ∙ 𝜙(𝑦)
𝜙 𝑥 + 𝑦 = 𝜙 𝑥 + 𝜙(𝑦)
• Isomorphism
one-to-one (1-1)
Onto- a function such that every element in B is
mapped by some element A
Homomorphism
Automorphis
m A map of ϕ: 𝐺 → 𝐺
is said to be
automorphism if ϕ
is 1-1, onto, and
homomorphism.

5. EXAMPLE
(ℂ, +)
𝜙: ℂ → ℂ
𝜙 𝑎 + 𝑏𝑖 = 𝑎 − 𝑏𝑖
𝜙 𝑧1 = 𝜙 𝑧2
⇒ 𝜙 𝑎1 + 𝑏1𝑖 = 𝜙 𝑎2 − 𝑏2𝑖
⇒ 𝑎1 + 𝑏1𝑖 = 𝑎2 − 𝑏2𝑖
⇒ 𝑎1 = 𝑎2 𝑎𝑛𝑑 𝑏1 = 𝑏2
⇒ 𝑎1 + 𝑏1𝑖 = 𝑎2 + 𝑏2𝑖
⇒ 𝑧1 = 𝑧2
∴ 𝝓 𝒊𝒔 𝒐𝒏𝒆 − 𝒕𝒐 − 𝒐𝒏𝒆
𝑧 = 𝑎 + 𝑏𝑖 ∈ ℂ
𝑧′ = 𝑎 − 𝑏𝑖 ∈ ℂ
⇒ 𝜙 𝑧′ = 𝜙(𝑎 − 𝑏𝑖)
= 𝑎 + 𝑏𝑖 = z
∴ 𝝓 𝒊𝒔 𝒐𝒏𝒕𝒐
𝜙 𝑧1 + 𝑧2 = 𝜙 𝑎1 + 𝑎2 + 𝑖 𝑏1 + 𝑏2
= 𝑎1 + 𝑎2 − 𝑖(𝑏1 + 𝑏2)
= 𝑎1 − 𝑏1𝑖 + 𝑎2 − 𝑏2𝑖
= 𝜙 𝑎1 + 𝑏1𝑖 + 𝜙 𝑎2 + 𝑏2𝑖
= 𝜙 𝑧1) + 𝜙(𝑧2
∴ 𝝓 𝒊𝒔 𝒉𝒐𝒎𝒐𝒎𝒐𝒓𝒑𝒉𝒊𝒔𝒎
𝒐𝒏𝒆 − 𝒕𝒐 − 𝒐𝒏𝒆
𝒐𝒏𝒕𝒐
𝒉𝒐𝒎𝒐𝒎𝒐𝒓𝒑𝒉𝒊𝒔𝒎
AUTOMORPHISM

6. 𝛼 ∈ 𝐴𝑢𝑡 ℤ10
𝛼: ℤ10 → ℤ10
𝛼 1
𝛼 𝑘
𝛼 𝑘 =
𝛼 1+1+⋯+1
𝑘 𝑡𝑒𝑟𝑚𝑠
=
𝛼 1 +𝛼 1 +⋯+𝛼 1
𝑘 𝑡𝑒𝑟𝑚𝑠
= 𝜶 𝟏 𝒌
𝛼 1 =𝐴𝑢𝑡(ℤ10)
Theorem 6.2 Properties of Isomorphic Acting on
Elements
Property 4. 𝐺 = 1 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝐺 = 𝛼(1)
⇒ ℤ10 = 1 ⇔ ℤ10 = 𝛼(1)
EXAMPLE 𝑨𝒖𝒕(ℤ𝟏𝟎)
Generators of ℤ10 are relatively prime to it:
𝟏, 𝟑, 𝟕, 𝟗
𝛼 1 𝑘
1 → 𝛼1
3 → 𝛼3
7 → 𝛼7
9 → 𝛼9

7. 𝑨𝒖𝒕 ℤ𝟏𝟎 = 𝛼1, 𝛼3, 𝛼7, 𝛼9
𝛼1 → 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦
𝛼3
𝛼3 𝑥 = 𝛼3 𝑦
⇒ 3x(mod 10) = 3y (mod 10)
⇒ x(mod 10) = y (mod 10)
𝛼3 1 = 3
3 is generator of ℤ𝟏𝟎
𝒐𝒏𝒆 − 𝒕𝒐 − 𝒐𝒏𝒆
𝒐𝒏𝒕𝒐
𝛼3 𝑎 + 𝑏 = 3 𝑎 + 𝑏
= 𝛼3 𝑎) + 𝛼3(𝑏
𝒉𝒐𝒎𝒐𝒎𝒐𝒓𝒑𝒉𝒊𝒔𝒎
3x mod 10
𝛼3 1 →
𝛼3 3 →
𝛼3 7 →
𝛼3 9 →
3
9
1
7
𝛼3 𝑎 + 𝑏 = 𝛼3 𝑎) + 𝛼3(𝑏
𝛼3 1 + 2 = 𝛼3 1) + 𝛼3(2
= 3 1) + 3(2
𝑜𝑟
= 3 + 6
𝛼3 3 → 9

8. Uses of
Automorphisms
• Automorphisms of groups can be used as a means of
constructing new groups from the original group.
• We can use the automorphisms group machinery to
determine the characteristic subgroups of a group.
• In computer science, automorphisms are useful in
understanding the complexity of algebraic problems.