JustToThePoint English Website Version
JustToThePoint en español
JustToThePoint in Thai

Isomorphisms. Cayley's Theorem.

Don’t worry if people think you’re crazy. You are crazy. You have that kind of intoxicating insanity that lets other people dream outside of the lines and become who they’re destined to be, Jennifer Elisabeth, Born Ready: Unleash Your Inner Dream Girl.

Let (A, *) and (B, ⋄) be two binary algebraic structures. A homomorphism is a structure-preserving map between two algebraic structures of the same type (groups, rings, fields, vector spaces, etc.) or, in other words, a map ϕ: A → B, such that ∀x, y ∈ S : ϕ(x ∗ y) = ϕ(x) ⋄ ϕ(y).

Notice that ϕ may not be one to one (injection), nor onto (surjection). An isomorphism is a bijective homomorphism, i.e., one-to-one and onto. In other words, let (S, *) and (S’, ⋄) be two binary algebraic structures of the same type. An isomorphism of S with S' is a 1-1 function ϕ mapping from S onto S′ such that the homomorphism property holds: ∀x, y ∈ S : ϕ(x ∗ y) = ϕ(x) ⋄ ϕ(y). S and S’ are said to be isomorphic binary structures and we denote or write it by S ≃ S’ or S ≋ S'.

In particular, two groups (G, *) and (G', ⋄) are isomorphic if there exist a bijective homomorphism, i.e., a one-to-one and onto map Φ: G → G’ such that the group operation is preserved, that is, ∀x, y ∈ G : ϕ(x ∗ y) = ϕ(x) ⋄ ϕ(y). Isomorphic groups are equivalent, that is, they share the same algebraic and group theoretic properties.

Examples

  1. First, we define the function ϕ: ℤ → 2ℤ, ϕ(n) = 2n.
  2. Let’s see that ϕ is a one-to-one function. ϕ(n)=ϕ(m) ⇒ 2n = 2m ⇒ n = m.
  3. ϕ is onto S’. ∀n ∈ 2ℤ, then ∃m ∈ ℤ: n = 2m, ϕ(m) = 2m = n.
  4. ϕ(m+n) = 2(m+n) = 2m + 2n = ϕ(m) + ϕ(n).
  1. First, we define the function ϕ: ℝ → ℝ+. ϕ(x) = ex.
  2. Let’s see that ϕ is a one-to-one function. ϕ(n)=ϕ(m) ⇒ en = em ⇒ ln(en) = ln(em) ⇒ n = m.
  3. ϕ is onto S’. ∀r’ ∈ ℝ+, ∃r ∈ ℝ such that r = ln(r’), and obviously ϕ(r) = er = eln(r’) = r'.
  4. ϕ(r+s) = er+s = eres = ϕ(r)ϕ(s).
  1. First, we define the function ϕ: ℝ+ → ℝ. ϕ(r) = ln(r).
  2. Let’s see that ϕ is a one-to-one function. ϕ(n)=ϕ(m) ⇒ ln(n) = ln(m) ⇒ eln(n) = eln(m) ⇒ n = m.
  3. ϕ is onto S’. ∀r ∈ ℝ, ∃r’ ∈ ℝ+ such that r’= er, and obviously ϕ(r’) = ln(er) = r.
  4. ϕ(r+s) = ln(r+s) = ln(r) + ln(s) = ϕ(r) + ϕ(s).

There is an homomorphism between ℂ* and $\Tau$, ϕ: ℂ* → $\Tau$, ϕ(re) = e.

Let z = re, w = se ⇒ ϕ(zw) = ei(α+β) = (e)(e) = ϕ(z)ϕ(w), but it is not obviously onto because for every point in the unit circle, there are infinite number of complex numbers which map to it.

However, there is an isomorphism between U = $\Tau$ and ℝ with addition modulo 2π. z1 = e1 (z1 ↔ θ1), z2 = e2 (z1 ↔ θ1) then z1 + z2 ↔ θ1 + θ2.

In other words, the algebraic properties of U and ℝ are the same, e.g., z4 = 1 has four solutions in U, namely 1, i, -1, -i, and x + + x + + x + x = 0 has four solutions in ℝ (0, π, π/2, 3π/2), too.

Proof by contradiction: Suppose that Φ were such a mapping, there would be an a ∈ ℚ: Φ(a) = -1 ⇒ -1 = Φ(a) = Φ(a/2 + a/2) = Φ(a/2)Φ(a/2) = Φ(a/2)2 ⇒ there would be an a ∈ ℚ: -1 = Φ(a/2)2

U12 = {1, 5, 7, 11}. Assume there is an isomorphism Φ: U10 → U12, Φ(9) = Φ(3·3) = Φ(3)Φ(3) = (Φ(3))2 = [Observe that x2 = 1 ∀x ∈ U12] 1. Similarly, Φ(1) = Φ(1·1) = Φ(1)Φ(1) = (Φ(1))2 = 1. Therefore, Φ(9) = Φ(1), that is, Φ is not one to one, ⊥

  1. ΦM(A) = MAM-1 ∈ SL(2, ℝ)? det(MAM-1) = (det M)(det A)(det M)-1 = 1 · 1 · 1 = 1.
  2. ΦM(A) = ΦM(B) ⇒ MAM-1 = MBM-1 ⇒ [∴ SL(2, ℝ) is a group and cancellation laws do apply] A = B.
  3. B ∈ SL(2, ℝ), we must find a matrix A: ΦM(A) = MAM-1 = B, but this is really trivial, it is indeed A = M-1BM.
  4. ΦM(AB) = M(AB)M-1 = MA(M-1M)BM-1 = (MAM-1)(MBM-1) = ΦM(A)ΦM(B).
  1. Let’s define Φ :D6 → S3, Φ(r) = a, Φ(s) = b. It is an homomorphism.
  2. S3 is generated by a and b (it is left as an exercise) ⇒ Φ is surjective.
  3. |D6| = |S3| = 6 ⇒ Φ is isomorphism, D6 ≋ S3.

Cayley’s Theorem

Cayley’s Theorem. Every group G is isomorphic to a subgroup of a symmetric group. In particular, every finite group is isomorphic to a subgroup of Sn.

Recall: The symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group Sn defined over a finite set of n symbols consists of all the permutations that can be performed on the n symbols.

Proof.

Theorem. A function is bijective if and only if has an inverse. Therefore, fg is bijective ⇒ fg is a permutation of G ⇒ fg ∈ Sym(G).

  1. Associative. The operation of function composition is associative.
  2. Close under function composition. ∀g, h, x ∈ G: fg·fh(x) = fg(fh(x)) = fg(h·x) = g·(h·x) = [G group. Associative] (g·h)·x = fgh(x), gh ∈ G (G is a group, ∀g, h, gh ∈ G) ⇒ fgh ∈ K.
  3. fe is the identity.
  4. ∀g ∈ G, (fg)-1 = fg-1 (it was previously proven).
  1. First, we define the function Φ: G → K, Φ(g) = fg.

  2. Φ is a group homomorphism, Φ(g·h) = fg·h = [*1] fg·fh = Φ(g)·Φ(h)

    [*1]fg·fh(x) = fg(fh(x)) = fg(h·x) = g·(h·x) =[(G, ·) group ⇒ Associativity] (g·h)·x = fg·h(x) ∀x ∈G

  3. Φ(g) = Φ(h) ⇒ fg = fh ⇒ ∀x ∈ G, fg(x) = fh(x) ⇒ ∀x ∈ G, g·x = h·x. The identity element, e ∈ G ⇒ g·e = h·e ⇒ g = h ⇒ Φ is one-to-one.

  4. By the way K = {fg: g ∈ G} was constructed, Φ is obviously onto

Definition. The map Φ in the proof of Cayley’s Theorem is the left representation of G.

Examples of the regular group representation.

Bibliography

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. This post relies heavily on the following resources, specially on NPTEL-NOC IITM, Introduction to Galois Theory, Michael Penn, and Contemporary Abstract Algebra, Joseph, A. Gallian.
  1. NPTEL-NOC IITM, Introduction to Galois Theory.
  2. Algebra, Second Edition, by Michael Artin.
  3. LibreTexts, Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
  4. Field and Galois Theory, by Patrick Morandi. Springer.
  5. Michael Penn (Abstract Algebra), and MathMajor.
  6. Contemporary Abstract Algebra, Joseph, A. Gallian.
  7. Andrew Misseldine: College Algebra and Abstract Algebra.
Bitcoin donation

JustToThePoint Copyright © 2011 - 2024 Anawim. ALL RIGHTS RESERVED. Bilingual e-books, articles, and videos to help your child and your entire family succeed, develop a healthy lifestyle, and have a lot of fun.

This website uses cookies to improve your navigation experience.
By continuing, you are consenting to our use of cookies, in accordance with our Cookies Policy and Website Terms and Conditions of use.