No pressure, no diamonds, Thomas Carlyle.
For every problem there is always, at least, a solution which seems quite plausible. It is simple and clean, direct, neat and nice, and yet very wrong, #Anawim, justtothepoint.com
Recall:
(13)(12)(13) = (23), notice that (13)^{-1} = (13). (23)(12)(23) = (13)
Theorem. For each cycle (i_{1}i_{2}···i_{k}) in S_{n} and each σ ∈ S_{n}, σ(i_{1}i_{2}···i_{k})σ^{-1} = (σ(i_{1})σ(i_{2})···σ(i_{k}))
Proof.
Let π = σ(i_{1}i_{2}···i_{k})σ^{-1}.
π(σ(i_{1})) = σ(i_{1}i_{2}···i_{k})σ^{-1}σ(i_{1}) = σ(i_{1}i_{2}···i_{k})(σ^{-1}σ)(i_{1}) = σ(i_{1}i_{2}···i_{k})(i_{1}) = σ(i_{2}). Therefore, π sends σ(i_{1}) to σ(i_{2}), σ(i_{2}) to σ(i_{3}), …, σ(i_{k}) to σ(i_{1})
Let’s show that π does not move anything other that σ(i_{1}), σ(i_{2})···σ(i_{k}). Let pick a that is not among σ(i_{1}), σ(i_{2})···σ(i_{k}). Claim: π(a) = a ↭ σ(i_{1}i_{2}···i_{k})σ^{-1}(a) = a.
Since a ≠ σ(i_{j}) for j = 1, 2, ···, k ⇒ σ^{-1}(a) ≠ i_{j} for j = 1, 2, ···, k ⇒ The cycle (i_{1}i_{2}···i_{k}) does not move it (“it” means σ^{-1}(a)) ⇒ π(a) = σ(i_{1}i_{2}···i_{k})σ^{-1}(a) = σ(i_{1}i_{2}···i_{k})(σ^{-1}(a)) = σ(σ^{-1}(a)) = a∎
Theorem. All cycles of the same length in S_{n} are conjugate.
Proof.
Let’s pick two cycles of the same length in S_{n}, say (a_{1}a_{2}···a_{k}), (b_{1}b_{2}···b_{k}). Let’s select σ ∈ S_{n} such that σ(a_{1}) = b_{1}, σ(a_{2}) = b_{2}, ··· σ(a_{k}) = b_{k}, and let σ be an arbitrary bijection from the complement of {a_{1}, a_{2}, ···, a_{k}} to the complement of {b_{1}, b_{2}, ···, b_{k}}.
Then, using the previous theorem, we see conjugation by σ carries the first k-cycle to the second.
Lemma 1. For all n ≥ 3, the alternating group A_{n} is generated by 3 cycles.
Proof.
The identity (1) is (123)(132), which is a product of 3-cycles. Now, let’s take a non-identity element of A_{n}, say σ and write it as a product of transpositions, σ = τ_{1}τ_{2}···τ_{r}.
By definition, A_{n} = {σ ∈ S_{n} : σ is even, sgn(σ) = 1}, so the left side has sign 1 (Recall: sgn(σ) = (-1)^{m} where m is the number of transpositions in the decomposition of σ), and the right side has sign (-1)^{r}, so r is even. Therefore, we can collect the products on the right side of the equation into successive transpositions τ_{i}τ_{i+1}, i = 1, 3, and so on.
We will now show that every product of two transpositions in S_{n} is a product of two 3-cycles, hence σ is a product of 3-cycles. There are three possibilities:
Lemma 2. For n ≥ 5, all 3-cycles are conjugate in A_{n}
Proof.
We will show every 3-cycle in A_{n} is conjugate within A_{n} to (123). We are not really losing any generality over here.
Let σ be a 3-cycle in A_{n} ⇒ [Previous Theorem. All cycles of the same length in S_{n} are conjugate. This may no be true in subgroups of S_{n}] σ can be conjugated to (123) in S_{n}: ∃τ ∈S_{n} such that τστ^{-1} = (123)
If τ ∈ A^{n}, we are done, 🎉
Otherwise, τ ∉ A_{n}, let τ’ = (45)τ, so τ’∈ A_{n} and τ’στ’^{-1} = [(45)^{-1} = (45), Shoes and sock Theorem] (45)τστ^{-1}(45) = [τστ^{-1} = (123)] (45)(123)(45) = (123) ∎
Theorem. A_{5} is simple.
A_{n} is simple ∀n ≥ 5, and S_{n}, A_{n} are not solvable ∀n ≥ 5.
Proof
Let’s suppose for the sake of contradiction that there exist N ◁ A_{5}, a non-trivial normal subgroup of A_{5}, |N| > 1. So we can pick σ ∈ N, σ ≠ (1).
Claim: N = A_{5}. We will show that N contains a 3 cycle ⇒ [Lemma 2 and N ◁ A_{5}] all 3-cycles are conjugate in A_{5} (n = 5) and they are all contained in N (normal) ⇒ [Lemma 1] N = A_{5}.
N ⊆ A_{5}, so only even permutations are allowed in A_{5}, that is, the cycle structure of σ is a three-cycle, say (abc) (🎉), (ab)(cd), and (abcde)
(ab)(cd) ∈ N ⇒ [N ◁ A_{5}] (abe)(ab)(cd)(abe)^{-1} ∈ N, then (abe)(ab)(cd)(abe)^{-1}(ab)(cd) ∈ N, but (abe)(ab)(cd)(abe)^{-1}(ab)(cd) = (abe)(ab)(cd)(aeb)(ab)(cd) = (aeb) ∈ N
(abcde) ∈ N ⇒ [N ◁ A_{5}] (abc)(abcde)(abc)^{-1}∈ N ⇒ [N is a subgroup, too] ((abc)(abcde)(abc)^{-1})(abcde)^{-1} = ((abc)(abcde)(acb))(aedcb) = (adebc)(aedcb) = (abd)∈ N
Definition. An extension of fields K/F is called simple if K = F(α) for some element α. In this case, α is called a primitive element for K.
Primitive element theorem. Let K/F be a finite extension. Then, a finite extension of fields is primitive if and only if there exists only finitely many subfields of K containing F.
Proof:
⇒ Let K/F be a primitive extension, K = F(α), and let f ∈ F[x] be the irreducible polynomial of α over F. There is an injective map from the intermediate fields of K/F to the set of divisors of f in K[x] which is obviously a finite set, Φ: {intermediate fields of K/F} → {divisors of f in K[x]}
Let L be an intermediate field, and let g ∈ L[x] be the irreducible polynomial of α over L. Hence, g divides f in L[x]
Let L’ be the field generated over F by the coefficients of g(x). Obviously, g ∈ L’[x], and since K = F(α), then K = L(α) = L’(α).
[K : L] = degree of the irreducible polynomial of α over L = deg(g) = degree of the irreducible polynomial of α over L’ = [K : L’] ⇒ [K : L] = [K : L’] and L’⊆ L ⇒ L = L’ = F(coefficients of g), g | f in L[x] and K[x]. It follows that the subfields of K containing F are the subfields generated by the coefficients of the monic factors of f(x), hence there are only finitely many such subfields. This is one reasoning, let’s see the same result with another argument.
The map is defined as follows, Φ: {intermediate fields of K/F} → {divisors of f in K[x]}, Φ(L) = g where g is the irreducible polynomial of α over L, g divides f in L[x], and obviously in K[x], too.
Suppose Φ(L_{1}) = Φ(L_{2}) = g ⇒ L_{1} = F(coefficients of g) = L_{2} ⇒ Φ is one to one ⇒ Since {divisors of f in K[x]} is a finite set, K/F has only finitely many intermediate fields.
⇐ Suppose conversely that K/F has only finitely many intermediate fields.
It suffices to show that ∀α, β ∈ K, ∃ γ ∈ K such that F(α, β) = F(γ)
Case base, n = 2, F(α, β) = F(γ). Induction hypothesis, F(α_{1}, α_{2}, ···, α_{n-1}) = F(γ’), therefore F(α_{1}, α_{2}, ···, α_{n}) = F(γ’, α_{n}) = [n = 2] F(γ’’)
∀c ∈ F, consider α + cβ ∈ F(α, β), and the subfields F(α +cβ), F ⊆ F(α + cβ) ⊆ F(α, β) ⊆ K. Therefore, since there are obviously infinitely many choices for c ∈ F and (by assumption) only finitely many such subfields ⇒ ∃c_{1}, c_{2} ∈ F, c_{1} ≠ c_{2}, such that F(α + c_{1}β) = F(α + c_{2}β).
Futhermore, α + c_{1}β and α + c_{2}β both lie in F(α + c_{1}β), and taking their differences show that (α + c_{1}β) - (α + c_{2}β) = (c_{1} -c_{2})β ∈ F(α + c_{1}β) ⇒ [c_{1}≠c_{2}, so c_{1}-c_{2}≠0] β ∈ F(α + c_{1}β), and then [c_{1}β ∈ F(α + c_{1}β)] also α = α + c_{1}β -c_{1}β ∈ F(α + c_{1}β) ⇒ F(α, β) ⊆ F(α + c_{1}β) ⊆_{trivially} F(α, β) ⇒ F(α, β) = F(α + c_{1}β)∎
Corollary. A finite separable extension K/F is primitive.
Proof.
Let L be the Galois closure of K over F. Since L/F is Galois, by Galois’ Main Theorem, L/F has only finitely many intermediate fields (there is a bijection between these intermediate fields and the subgroups of its Galois group which is indeed a finite set) ⇒ [F ⊆ K ⊆ L] K/F has only finitely many intermediate fields ⇒ [Primitive element theorem. Let K/F be a finite extension. Then, a finite extension of fields is primitive if and only if there exists only finitely many subfields of K containing F.] K/F is primitive ∎
Corollary. Any finite extension of fields of characteristic zero is simple.
Proof.
It follows from the previous corollary since any finite extension of fields in characteristic zero is separable∎