Lagrange

GaloisÉvariste GaloisandSolvable Permutation Groups
David A. Cox
Department of Mathematics
Amherst College
dac@math.amherst.edu

BilbaoMay 2012

Solvable Permutation Groups

Lagrange

GaloisSolvable Permutation Groups

Prologue

Most mathematicians know about Galois:
He introduced the concept of group.
He created Galois theory.
He defined finite fields.
This Talk
We will learn some of the amazing other things Galois did,especially his work on solvable permutation groups.

Lagrange

GaloisSolvable Permutation Groups

Prologue

Most mathematicians know about Galois:
He introduced the concept of group.
He created Galois theory.
He defined finite fields.
This Talk
We will learn some of the amazing other things Galois did,especially his work on solvable permutation groups.

Lagrange

GaloisSolvable Permutation Groups

Prologue

Most mathematicians know about Galois:
He introduced the concept of group.
He created Galois theory.
He defined finite fields.
This Talk
We will learn some of the amazing other things Galois did,especially his work on solvable permutation groups.

Lagrange

GaloisSolvable Permutation Groups

Prologue

Most mathematicians know about Galois:
He introduced the concept of group.
He created Galois theory.
He defined finite fields.
This Talk
We will learn some of the amazing other things Galois did,especially his work on solvable permutation groups.

Lagrange

GaloisOutline1

Lagrange
Three Snapshots2

GaloisFour Snapshots

3

Solvable Permutation Groups
Primitive Equations
The Affine Linear Group
Finite Fields
The Main Theorem

Solvable Permutation Groups

Lagrange

GaloisOutline1

Lagrange
Three Snapshots2

GaloisFour Snapshots

3

Solvable Permutation Groups
Primitive Equations
The Affine Linear Group
Finite Fields
The Main Theorem

Solvable Permutation Groups

Lagrange

GaloisOutline1

Lagrange
Three Snapshots2

GaloisFour Snapshots

3

Solvable Permutation Groups
Primitive Equations
The Affine Linear Group
Finite Fields
The Main Theorem

Solvable Permutation Groups

Lagrange

GaloisSolvable Permutation Groups

Three Snapshots from Lagrange

In 1770, Lagrange wrote the wonderful paper:
Réflexions sur la résolution des équations
which introduced many of the key players of group theory andGalois theory.
We will give three excerpts from this paper to show how
Lagrange laid the foundation for what Galois did.

Lagrange

GaloisSolvable Permutation Groups

Three Snapshots from Lagrange

In 1770, Lagrange wrote the wonderful paper:
Réflexions sur la résolution des équations
which introduced many of the key players of group theory andGalois theory.
We will give three excerpts from this paper to show how
Lagrange laid the foundation for what Galois did.

Lagrange

GaloisSolvable Permutation Groups

Three Snapshots

Snapshot #1
A QuoteA function y of the roots of a polynomial gives values y ′ , y ′′ , . . .
when the roots are permuted. Lagrange says:
il s’agit ici uniquement de la forme de valeurs et non de leurquantité absolue.

Since y =A(α1 , . . . , αn )A(x1 , . . . , xn )has the form,
B(α1 , . . . , αn )B(x1 , . . . , xn )we see that Lagrange regarded the roots as variables!

Lagrange

GaloisSolvable Permutation Groups

Three Snapshots

Snapshot #2
The Universal Extension
Let σi be the ith elementary symmetric polynomial. Consider
K = k(σ1 , . . . , σn ) ⊆ L = k(x1 , . . . , xn ).
Lagrange’s Theorem
If t, y ∈ L and t is fixed by all permutations that fix y, then t is arational expression in y, σ1 , . . . , σn .
K ⊆ K (y) ⊆ L gives Gal(L/K (y)) = {permutations fixing y}. Sot ∈ fixed field of Gal(L/K (y)) =⇒ t ∈ K (y).
Hence K (y) is the fixed field of Gal(L/K (y)).
Lagrange knew the Galois correspondence!

Lagrange

GaloisSolvable Permutation Groups

Three Snapshots

Snapshot #2
The Universal Extension
Let σi be the ith elementary symmetric polynomial. Consider
K = k(σ1 , . . . , σn ) ⊆ L = k(x1 , . . . , xn ).
Lagrange’s Theorem
If t, y ∈ L and t is fixed by all permutations that fix y, then t is arational expression in y, σ1 , . . . , σn .
K ⊆ K (y) ⊆ L gives Gal(L/K (y)) = {permutations fixing y}. Sot ∈ fixed field of Gal(L/K (y)) =⇒ t ∈ K (y).
Hence K (y) is the fixed field of Gal(L/K (y)).
Lagrange knew the Galois correspondence!

Lagrange

GaloisSolvable Permutation Groups

Three Snapshots

Snapshot #2
The Universal Extension
Let σi be the ith elementary symmetric polynomial. Consider
K = k(σ1 , . . . , σn ) ⊆ L = k(x1 , . . . , xn ).
Lagrange’s Theorem
If t, y ∈ L and t is fixed by all permutations that fix y, then t is arational expression in y, σ1 , . . . , σn .
K ⊆ K (y) ⊆ L gives Gal(L/K (y)) = {permutations fixing y}. Sot ∈ fixed field of Gal(L/K (y)) =⇒ t ∈ K (y).
Hence K (y) is the fixed field of Gal(L/K (y)).
Lagrange knew the Galois correspondence!

Lagrange

GaloisSolvable Permutation Groups

Three Snapshots

Snapshot #3
Definition
Rational functions t, y ∈ L are similar (“semblable”) when apermutation fixes t if and only if it fixes y.
Corollary of Lagrange’s Theorem
Functions t, y ∈ L are similar if and only if
K (t) = K (y).
For us, the Galois correspondence of K ⊆ L is a bijection
intermediate fields ←→ subgroups of Sn .
Lagrange worked with individual rational functions. The notionof “similar function” is his attempt at an intrinsic formulation.
Lagrange knew the Galois correspondence!

Lagrange

GaloisSolvable Permutation Groups

Three Snapshots

Snapshot #3
Definition
Rational functions t, y ∈ L are similar (“semblable”) when apermutation fixes t if and only if it fixes y.
Corollary of Lagrange’s Theorem
Functions t, y ∈ L are similar if and only if
K (t) = K (y).
For us, the Galois correspondence of K ⊆ L is a bijection
intermediate fields ←→ subgroups of Sn .
Lagrange worked with individual rational functions. The notionof “similar function” is his attempt at an intrinsic formulation.
Lagrange knew the Galois correspondence!

Lagrange

GaloisSolvable Permutation Groups

Three Snapshots

Snapshot #3
Definition
Rational functions t, y ∈ L are similar (“semblable”) when apermutation fixes t if and only if it fixes y.
Corollary of Lagrange’s Theorem
Functions t, y ∈ L are similar if and only if
K (t) = K (y).
For us, the Galois correspondence of K ⊆ L is a bijection
intermediate fields ←→ subgroups of Sn .
Lagrange worked with individual rational functions. The notionof “similar function” is his attempt at an intrinsic formulation.
Lagrange knew the Galois correspondence!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots from Galois

Évariste Galois was born October 25, 1811 and died May 31,
1832. We are celebrating the 200th anniversary of his birth.
In January 1831, he wrote the amazing paper:
Mémoire sur les conditions de résolubilité des équations
par radicaux
I will give four excerpts from this memoir to give you a sense of
how Galois thought about Galois theory.

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #1
P ROPOSITION I
T HÉORÈME. Soit une équation donnée, dont a,b,c,. . . sont les
m racines. Il y aura toujours un groupe de permutations deslettres a,b,c,. . . qui jouira des propriété suivant:1o que toute fonction des racines, invariable∗∗ par les
substitutions de ce groupe, soit rationallement connue;
2 o réciproquement, que toute fonction des racines
déterminable rationallement, soit invariable par ces
substitutions∗ .
The asterisks ∗∗ and ∗ indicate marginal notes in Galois’smanuscript.

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #1
Marginal Note for “invariable∗∗ ”
Nous appelons ici invariable non seulement une fonction dont
la forme est invariable par les substitutions des racines entre
elles, mais encore celle dont las valeur numérique ne varieriatpas par ces substitutions.
Galois is aware that his theory applies to the roots of any
polynomial, not just the case when the roots are variables.
Galois has gone beyond Lagrange!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #2
Galois’s Group
Quelle que soit l’équation donnée, on pourra trouver une
fonction V des racines telle que toutes les racines soient
fonctions rationelles des V . Cela posé, considérons l’equation
irréductible done V est racine (lemmes III et IV). Soient
V , V ′ , V ′′ , . . . , V (n−1) les racines de cette equation.
Soient ϕV , ϕ1 V , ϕ2 V , . . . , ϕm−1 V les racines de la proposée.
Écrivons les n permutations suivants des racines:
(V ),
ϕV,
ϕ1 V ,
ϕ2 V ,...,
ϕn−1 V ,
′
′
′
′
(V ),
ϕV ,
ϕ1 V ,
ϕ2 V ,
...,
...,
...,
...,
...,
...,
...,
...,
(m−1)
(m−1)
(m−1)
(m−1)
(V
), ϕ V, ϕ1 V, ϕ2 V, . . . , ϕn−1 V (m−1)
Je dis que ce groupe de permutations jouit de la propriétéénoncée.

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #2
For Galois, permutations are bijections{1, . . . , m} −→ {roots} ⇐ call these arrangements

while substitutions are bijections{roots} −→ {roots} ⇐ call these substitutions.
Arrangements give a strong visual picture.
Substitutions give a group.
Arrangements give a principal homogeneous space for thesubstitutions.
Marginal note for “substitutions∗ ”
Mettre partout á la place du mot permutation le mot substitution
But then he crosses this out!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #2
For Galois, permutations are bijections{1, . . . , m} −→ {roots} ⇐ call these arrangements

while substitutions are bijections{roots} −→ {roots} ⇐ call these substitutions.
Arrangements give a strong visual picture.
Substitutions give a group.
Arrangements give a principal homogeneous space for thesubstitutions.
Marginal note for “substitutions∗ ”
Mettre partout á la place du mot permutation le mot substitution
But then he crosses this out!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #2
For Galois, permutations are bijections{1, . . . , m} −→ {roots} ⇐ call these arrangements

while substitutions are bijections{roots} −→ {roots} ⇐ call these substitutions.
Arrangements give a strong visual picture.
Substitutions give a group.
Arrangements give a principal homogeneous space for thesubstitutions.
Marginal note for “substitutions∗ ”
Mettre partout á la place du mot permutation le mot substitution
But then he crosses this out!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #2
For Galois, permutations are bijections{1, . . . , m} −→ {roots} ⇐ call these arrangements

while substitutions are bijections{roots} −→ {roots} ⇐ call these substitutions.
Arrangements give a strong visual picture.
Substitutions give a group.
Arrangements give a principal homogeneous space for thesubstitutions.
Marginal note for “substitutions∗ ”
Mettre partout á la place du mot permutation le mot substitution
But then he crosses this out!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #2
For Galois, permutations are bijections{1, . . . , m} −→ {roots} ⇐ call these arrangements

while substitutions are bijections{roots} −→ {roots} ⇐ call these substitutions.
Arrangements give a strong visual picture.
Substitutions give a group.
Arrangements give a principal homogeneous space for thesubstitutions.
Marginal note for “substitutions∗ ”
Mettre partout á la place du mot permutation le mot substitution
But then he crosses this out!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #3
Galois introduced normal subgroups as follows.
P ROPOSITION III
T HÉORÈME. Si ll’on adjoint á une équation TOUTES les racinesd’une équation auxiliaire, les groupes dont il est question dansle théorème II jouiront de plus de cette propriété que lessubstitutions sont les mêmes dans chaque groupe.
If G is the Galois group and a is one arrangement, then G · a is
Galois’s “groupe.” The cosets of H ⊆ G giveG · a = g1 H · a ∪ g2 H · a ∪ · · · .
These are the “groupes” of P ROPOSITION III. Key observation:g(gi H · a) = gi H · a ⇐⇒ g ∈ gi Hgi−1 .
So “sont les mêmes” means H is normal!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #3
Galois introduced normal subgroups as follows.
P ROPOSITION III
T HÉORÈME. Si ll’on adjoint á une équation TOUTES les racinesd’une équation auxiliaire, les groupes dont il est question dansle théorème II jouiront de plus de cette propriété que lessubstitutions sont les mêmes dans chaque groupe.
If G is the Galois group and a is one arrangement, then G · a is
Galois’s “groupe.” The cosets of H ⊆ G giveG · a = g1 H · a ∪ g2 H · a ∪ · · · .
These are the “groupes” of P ROPOSITION III. Key observation:g(gi H · a) = gi H · a ⇐⇒ g ∈ gi Hgi−1 .
So “sont les mêmes” means H is normal!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #3
Galois introduced normal subgroups as follows.
P ROPOSITION III
T HÉORÈME. Si ll’on adjoint á une équation TOUTES les racinesd’une équation auxiliaire, les groupes dont il est question dansle théorème II jouiront de plus de cette propriété que lessubstitutions sont les mêmes dans chaque groupe.
If G is the Galois group and a is one arrangement, then G · a is
Galois’s “groupe.” The cosets of H ⊆ G giveG · a = g1 H · a ∪ g2 H · a ∪ · · · .
These are the “groupes” of P ROPOSITION III. Key observation:g(gi H · a) = gi H · a ⇐⇒ g ∈ gi Hgi−1 .
So “sont les mêmes” means H is normal!

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #4
Galois had a wonderful result about the Galois group of asolvable polynomial of prime degree.
P ROPOSITION VII
P ROBLÈME. Quel est le groupe d’une équation irréductible d’undegré premier n, soluble par radicaux?
Faisons en general xn = x0 , xn+1 = x1 , . . .
Donc, “si d’une équation irréductible de degré premier est
soluble par radicaux, le groupe de cette léquation ne saurait
contenir que les substitutions de la formexk
a et b étant des constants.”

xak +bLagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #4
In modern terms, Galois’s Proposition VII says that the Galois
group of an irreducible polynomial of prime p is solvable byradicals if and only if the Galois group is isomorphic to a
subgroup of the affine linear group
AGL(1, Fp ) = {x 7→ ax + b | a, b ∈ Fp , a 6= 0}.
There are two aspects of this result worth mentioning:
Up to conjugacy, AGL(1, Fp ) is the maximal solvable
subgroup of the symmetric group Sp .
This proposition implies that such a polynomial is solvableby radicals iff any two roots generate the splitting field.

Lagrange

GaloisSolvable Permutation Groups

Four Snapshots

Snapshot #4
In modern terms, Galois’s Proposition VII says that the Galois
group of an irreducible polynomial of prime p is solvable byradicals if and only if the Galois group is isomorphic to a
subgroup of the affine linear group
AGL(1, Fp ) = {x 7→ ax + b | a, b ∈ Fp , a 6= 0}.
There are two aspects of this result worth mentioning:
Up to conjugacy, AGL(1, Fp ) is the maximal solvable
subgroup of the symmetric group Sp .
This proposition implies that such a polynomial is solvableby radicals iff any two roots generate the splitting field.

Lagrange

GaloisSolvable Permutation Groups

There is More!

The four snapshots from Galois:
are well-known to most mathematicians, andillustrate nicely the power of the theory he developed.
However, Galois did a lot more!
In particular, we will explore:
what Galois knew about solvable permutation groups, andwhy he invented finite fields.

Lagrange

GaloisSolvable Permutation Groups

There is More!

The four snapshots from Galois:
are well-known to most mathematicians, andillustrate nicely the power of the theory he developed.
However, Galois did a lot more!
In particular, we will explore:
what Galois knew about solvable permutation groups, andwhy he invented finite fields.

Lagrange

GaloisSolvable Permutation Groups

There is More!

The four snapshots from Galois:
are well-known to most mathematicians, andillustrate nicely the power of the theory he developed.
However, Galois did a lot more!
In particular, we will explore:
what Galois knew about solvable permutation groups, andwhy he invented finite fields.

Lagrange

GaloisSolvable Permutation Groups

There is More!

The four snapshots from Galois:
are well-known to most mathematicians, andillustrate nicely the power of the theory he developed.
However, Galois did a lot more!
In particular, we will explore:
what Galois knew about solvable permutation groups, andwhy he invented finite fields.

Lagrange

GaloisSolvable Permutation Groups

There is More!

The four snapshots from Galois:
are well-known to most mathematicians, andillustrate nicely the power of the theory he developed.
However, Galois did a lot more!
In particular, we will explore:
what Galois knew about solvable permutation groups, andwhy he invented finite fields.

Lagrange

GaloisSolvable Permutation Groups

There is More!

The four snapshots from Galois:
are well-known to most mathematicians, andillustrate nicely the power of the theory he developed.
However, Galois did a lot more!
In particular, we will explore:
what Galois knew about solvable permutation groups, andwhy he invented finite fields.

Lagrange

GaloisSolvable Permutation Groups

Primitive Equations

The Main Problem
Definition
On appelle équations non primitives les équations qui étant,par exemple, du degré mn, se décomposent en m facteurs dudegré n, au moyen d’une seule équation du degré m.
Examplex 4 − 2 has degree 4 = 2 · 2. Adjoining roots of x 2 − 2 gives√ √ 
x4 − 2 = x2 − 2 x2 + 2 .
Thus x 4 − 2 is imprimitive (“non primitive”).
Revenons maintenant á notre object, et cherchons en général
dans quel cas une équation primitive est soluble par radicaux.

Lagrange

GaloisSolvable Permutation Groups

Primitive Equations

The Main Problem
Definition
On appelle équations non primitives les équations qui étant,par exemple, du degré mn, se décomposent en m facteurs dudegré n, au moyen d’une seule équation du degré m.
Examplex 4 − 2 has degree 4 = 2 · 2. Adjoining roots of x 2 − 2 gives√ √ 
x4 − 2 = x2 − 2 x2 + 2 .
Thus x 4 − 2 is imprimitive (“non primitive”).
Revenons maintenant á notre object, et cherchons en général
dans quel cas une équation primitive est soluble par radicaux.

Lagrange

GaloisSolvable Permutation Groups

Primitive Equations

The Main Problem
Definition
On appelle équations non primitives les équations qui étant,par exemple, du degré mn, se décomposent en m facteurs dudegré n, au moyen d’une seule équation du degré m.
Examplex 4 − 2 has degree 4 = 2 · 2. Adjoining roots of x 2 − 2 gives√ √ 
x4 − 2 = x2 − 2 x2 + 2 .
Thus x 4 − 2 is imprimitive (“non primitive”).
Revenons maintenant á notre object, et cherchons en général
dans quel cas une équation primitive est soluble par radicaux.

Lagrange

GaloisSolvable Permutation Groups

Primitive Equations

The First Result
Galois’s Version. . . pour qu’une équation primitive soit soluble par radicaux, it
faut que son degré soit de la forme p ν , p étant premier.
Definition
A subgroup G ⊆ Sn is imprimitive if{1, . . . , n} = R1 ∪ · · · ∪ Rk , k > 1, |Ri | > 1 for some iand elements of G preserve the Ri . Then G is primitive if it is
not imprimitive.
Theorem
If G ⊆ Sn is primitive and solvable, then n = p ν , p prime.

Lagrange

GaloisSolvable Permutation Groups

Primitive Equations

The First Result
Galois’s Version. . . pour qu’une équation primitive soit soluble par radicaux, it
faut que son degré soit de la forme p ν , p étant premier.
Definition
A subgroup G ⊆ Sn is imprimitive if{1, . . . , n} = R1 ∪ · · · ∪ Rk , k > 1, |Ri | > 1 for some iand elements of G preserve the Ri . Then G is primitive if it is
not imprimitive.
Theorem
If G ⊆ Sn is primitive and solvable, then n = p ν , p prime.

Lagrange

GaloisSolvable Permutation Groups

Primitive Equations

The First Result
Galois’s Version. . . pour qu’une équation primitive soit soluble par radicaux, it
faut que son degré soit de la forme p ν , p étant premier.
Definition
A subgroup G ⊆ Sn is imprimitive if{1, . . . , n} = R1 ∪ · · · ∪ Rk , k > 1, |Ri | > 1 for some iand elements of G preserve the Ri . Then G is primitive if it is
not imprimitive.
Theorem
If G ⊆ Sn is primitive and solvable, then n = p ν , p prime.

Lagrange

GaloisSolvable Permutation Groups

Primitive Equations

Proof
Assume G ⊆ Sn is primitive and solvable. To show: n = p ν .
Let N be a minimal normal subgroup of G. One can prove that
there is a simple group A such thatN ≃ Aν
This is a standard fact about minimal normal subgroups. Then:
G primitive =⇒ N transitive.
G solvable =⇒ N ≃ Fνp .

N transitive and abelian =⇒ its isotropy subgroups are all
equal and hence trivial. Thus:p ν = |N| = |orbit| · |isotropy subgroup| = n · 1 = n. QED

Lagrange

GaloisSolvable Permutation Groups

Primitive Equations

Proof
Assume G ⊆ Sn is primitive and solvable. To show: n = p ν .
Let N be a minimal normal subgroup of G. One can prove that
there is a simple group A such thatN ≃ Aν
This is a standard fact about minimal normal subgroups. Then:
G primitive =⇒ N transitive.
G solvable =⇒ N ≃ Fνp .

N transitive and abelian =⇒ its isotropy subgroups are all
equal and hence trivial. Thus:p ν = |N| = |orbit| · |isotropy subgroup| = n · 1 = n. QED

Lagrange

GaloisSolvable Permutation Groups

Primitive Equations

Proof
Assume G ⊆ Sn is primitive and solvable. To show: n = p ν .
Let N be a minimal normal subgroup of G. One can prove that
there is a simple group A such thatN ≃ Aν
This is a standard fact about minimal normal subgroups. Then:
G primitive =⇒ N transitive.
G solvable =⇒ N ≃ Fνp .

N transitive and abelian =⇒ its isotropy subgroups are all
equal and hence trivial. Thus:p ν = |N| = |orbit| · |isotropy subgroup| = n · 1 = n. QED

Lagrange

GaloisSolvable Permutation Groups

The Affine Linear GroupThe Affine Linear Group
When G is primitive and solvable, the proof just given showsthat N ≃ Fνp is normal in G. One can show without difficulty thatG ⊆ AGL(ν, Fp ) = {x 7→ Ax + b | A ∈ GL(ν, Fp ), b ∈ Fνp }.
Galois knew this!
Letter to Chevalier, 29 May 1832
Toutes les permuations d’une équation primitive soluble parradicaux sont de la forme
xk .l.m.../xak +bl+cm+...+f .a1 k +b1 l+c1 m+...+g. ...
k, l, m, . . . étant ν indices qui prenant chacun p valeurs
indiquent toutes les racines. Les indices sont pris suivant lemodule p, c’est-á-dire que la racines sera la même quand on
adjoutera á l’un des indices un multiple de p.

Lagrange

GaloisSolvable Permutation Groups

Finite Fields

Finite Fields and the Affine Semilinear Group
When a primitive of degree p ν is solvable by radicals, its Galois
group lies in
AGL(ν, Fp ) ⊆ Sp ν .
However, AGL(ν, Fp ) is not solvable when ν ≥ 2. What is its
maximal primitive solvable subgroup? This is the question
Galois wanted to answer.

Galois used the Galois theory of finite fields to create someprimitive solvable permutation subgroups of AGL(ν, Fp ):AGL(1, Fp ν ): The affine linear group over Fp ν is
{x 7→ ax + b | a, b ∈ Fp ν , a 6= 0}.
AΓL(1, Fp ν ): The affine semilinear group over Fp ν is
{x 7→ aσ(x) + b | a, b ∈ Fp ν , σ ∈ Gal(Fp ν /Fp ), a 6= 0}.

Lagrange

GaloisSolvable Permutation Groups

Finite Fields

Finite Fields and the Affine Semilinear Group
When a primitive of degree p ν is solvable by radicals, its Galois
group lies in
AGL(ν, Fp ) ⊆ Sp ν .
However, AGL(ν, Fp ) is not solvable when ν ≥ 2. What is its
maximal primitive solvable subgroup? This is the question
Galois wanted to answer.

Galois used the Galois theory of finite fields to create someprimitive solvable permutation subgroups of AGL(ν, Fp ):AGL(1, Fp ν ): The affine linear group over Fp ν is
{x 7→ ax + b | a, b ∈ Fp ν , a 6= 0}.
AΓL(1, Fp ν ): The affine semilinear group over Fp ν is
{x 7→ aσ(x) + b | a, b ∈ Fp ν , σ ∈ Gal(Fp ν /Fp ), a 6= 0}.

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

What Galois Knew
Why Do We Need Finite Fields?
C’est surtout dans la théorie des permutations . . . que laconsidération des racines imaginaires des congruences paraît
indispensable. Elle donne une moyen simple et facile dereconnaître dans quel cas une équation primitive soit soluble
par radicaux . . .
Galois notes that if the Galois group of a primitive equation liesin AΓL(1, Fp ν ), then the equation is solvable by radicals.
Galois Goes On To Say:
Cette remarque aurait peu d’importance, si je n’etais parvenu á
demontrer que réciproquement une équation primitive nesaurait soluble par radicaux, á moins de satisfaire aux
conditions que je viens d’enoncer.

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

What Galois Knew
Why Do We Need Finite Fields?
C’est surtout dans la théorie des permutations . . . que laconsidération des racines imaginaires des congruences paraît
indispensable. Elle donne une moyen simple et facile dereconnaître dans quel cas une équation primitive soit soluble
par radicaux . . .
Galois notes that if the Galois group of a primitive equation liesin AΓL(1, Fp ν ), then the equation is solvable by radicals.
Galois Goes On To Say:
Cette remarque aurait peu d’importance, si je n’etais parvenu á
demontrer que réciproquement une équation primitive nesaurait soluble par radicaux, á moins de satisfaire aux
conditions que je viens d’enoncer.

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

Subsequent Developments
Galois Knew That There Were Exceptions:J’excepte les équations du 9e et 25e degré.
In 1868, Jordan classifed primitive solvable subgroups of
Sp 2 . He used AΓL(1, Fp 2 ) together with two other groups.
Of Jordan’s groups, only AΓL(1, Fp 2 ) is doubly-transitive.
Doubly-transitive subgroups are automatically primitive,
though the converse (as shown by Jordan) is false.
Jordan’s result shows that Galois characterization of primitive
solvable equations has some gaps. But maybe Galois was
implicitly assuming double-transitivity. If this is the case, thenhis characterization (with the above exceptions) is remarkably
close to being complete.

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

Subsequent Developments
Galois Knew That There Were Exceptions:J’excepte les équations du 9e et 25e degré.
In 1868, Jordan classifed primitive solvable subgroups of
Sp 2 . He used AΓL(1, Fp 2 ) together with two other groups.
Of Jordan’s groups, only AΓL(1, Fp 2 ) is doubly-transitive.
Doubly-transitive subgroups are automatically primitive,
though the converse (as shown by Jordan) is false.
Jordan’s result shows that Galois characterization of primitive
solvable equations has some gaps. But maybe Galois was
implicitly assuming double-transitivity. If this is the case, thenhis characterization (with the above exceptions) is remarkably
close to being complete.

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

Subsequent Developments
Galois Knew That There Were Exceptions:J’excepte les équations du 9e et 25e degré.
In 1868, Jordan classifed primitive solvable subgroups of
Sp 2 . He used AΓL(1, Fp 2 ) together with two other groups.
Of Jordan’s groups, only AΓL(1, Fp 2 ) is doubly-transitive.
Doubly-transitive subgroups are automatically primitive,
though the converse (as shown by Jordan) is false.
Jordan’s result shows that Galois characterization of primitive
solvable equations has some gaps. But maybe Galois was
implicitly assuming double-transitivity. If this is the case, thenhis characterization (with the above exceptions) is remarkably
close to being complete.

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

Subsequent Developments
Galois Knew That There Were Exceptions:J’excepte les équations du 9e et 25e degré.
In 1868, Jordan classifed primitive solvable subgroups of
Sp 2 . He used AΓL(1, Fp 2 ) together with two other groups.
Of Jordan’s groups, only AΓL(1, Fp 2 ) is doubly-transitive.
Doubly-transitive subgroups are automatically primitive,
though the converse (as shown by Jordan) is false.
Jordan’s result shows that Galois characterization of primitive
solvable equations has some gaps. But maybe Galois was
implicitly assuming double-transitivity. If this is the case, thenhis characterization (with the above exceptions) is remarkably
close to being complete.

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

Subsequent Developments
Galois Knew That There Were Exceptions:J’excepte les équations du 9e et 25e degré.
In 1868, Jordan classifed primitive solvable subgroups of
Sp 2 . He used AΓL(1, Fp 2 ) together with two other groups.
Of Jordan’s groups, only AΓL(1, Fp 2 ) is doubly-transitive.
Doubly-transitive subgroups are automatically primitive,
though the converse (as shown by Jordan) is false.
Jordan’s result shows that Galois characterization of primitive
solvable equations has some gaps. But maybe Galois was
implicitly assuming double-transitivity. If this is the case, thenhis characterization (with the above exceptions) is remarkably
close to being complete.

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

Huppert’s Theorem
Here is the great theorem proved by Huppert in 1957.
TheoremAssume G ⊆ Sn is solvable and doubly-transitive. Then:n = p ν , p prime.
Furthermore, ifpν ∈/ {32 , 52 , 72 , 112 , 232 , 34 },then G ⊆ AΓL(1, Fp ν ) up to conjugacy.
Given that Galois invented groups in 1829, his level of insight isastonishing! So, in this 200th anniversary of Galois’s birth, wehave more to celebrate than we thought!

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

Huppert’s Theorem
Here is the great theorem proved by Huppert in 1957.
TheoremAssume G ⊆ Sn is solvable and doubly-transitive. Then:n = p ν , p prime.
Furthermore, ifpν ∈/ {32 , 52 , 72 , 112 , 232 , 34 },then G ⊆ AΓL(1, Fp ν ) up to conjugacy.
Given that Galois invented groups in 1829, his level of insight isastonishing! So, in this 200th anniversary of Galois’s birth, wehave more to celebrate than we thought!

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

Huppert’s Theorem
Here is the great theorem proved by Huppert in 1957.
TheoremAssume G ⊆ Sn is solvable and doubly-transitive. Then:n = p ν , p prime.
Furthermore, ifpν ∈/ {32 , 52 , 72 , 112 , 232 , 34 },then G ⊆ AΓL(1, Fp ν ) up to conjugacy.
Given that Galois invented groups in 1829, his level of insight isastonishing! So, in this 200th anniversary of Galois’s birth, wehave more to celebrate than we thought!

Lagrange

GaloisSolvable Permutation Groups

The Main Theorem

Huppert’s Theorem
Here is the great theorem proved by Huppert in 1957.
TheoremAssume G ⊆ Sn is solvable and doubly-transitive. Then:n = p ν , p prime.
Furthermore, ifpν ∈/ {32 , 52 , 72 , 112 , 232 , 34 },then G ⊆ AΓL(1, Fp ν ) up to conjugacy.
Given that Galois invented groups in 1829, his level of insight isastonishing! So, in this 200th anniversary of Galois’s birth, wehave more to celebrate than we thought!