simple groups and the classification problem (part...

Simple groups andthe ClassificationProblem (Part I)

Vipul Naik

Basic definitions

Prerequisites

Simplicity andfactorization

What are thesimple groups?

Initial observations

Simplicity proofs byinduction

The classificationof finite simplegroups

Statement of theresult

Simple classicalgroups

Classification: thehistory

Two sides to it

The constructive side

The negative side

Beauty andimportance of theclassification

Simple groups and the ClassificationProblem (Part I)

Vipul Naik

January 24, 2007


Vipul Naik

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Two sides to it


The negative side


Outline

Basic definitionsPrerequisitesSimplicity and factorization

What are the simple groups?Initial observationsSimplicity proofs by induction

The classification of finite simple groupsStatement of the resultSimple classical groups

Classification: the historyTwo sides to itThe constructive sideThe negative side

Beauty and importance of the classification


Vipul Naik

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What’s a group: a reminder

A group(defined) is a set G with a binary operation ∗ (calledmultiplication), a unary operation −1 (called inverse) and aconstant function e (called the identity element) satisfyingthe conditions of associativity, inverse, and multiplicativeidentity:

a ∗ (b ∗ c) = (a ∗ b) ∗ c ∀ a, b, c ∈ G (Associativity)

a ∗ e = e ∗ a = a ∀a ∈ G (multiplicative identity)

a ∗ a−1 = a−1 ∗ a = e ∀ a ∈ G (inverse)

Since ∗ is associative, we often omit both the ∗ symbol andthe parenthesization.


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Group homomorphisms and kernels

Let G and H be groups. A homomorphism ρ : G → H is aset- theoretic map such that ρ(g1g2) = ρ(g1)ρ(g2) for allg1, g2 ∈ G . Here, the multiplication on the left is in G andthe multiplication on the right is in H.

For any homomorphism ρ, the following are true:

ρ(g1g2) = ρ(g1)ρ(g2) ∀ g1, g2 ∈ G

ρ(g−1) = ρ(g)−1 ∀ g ∈ G

ρ(e) = e

The kernel of a homomorphism ρ : G → H is the inverseimage of the identity element of H via ρ.


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Group homomorphisms and kernels

Let G and H be groups. A homomorphism ρ : G → H is aset- theoretic map such that ρ(g1g2) = ρ(g1)ρ(g2) for allg1, g2 ∈ G . Here, the multiplication on the left is in G andthe multiplication on the right is in H.For any homomorphism ρ, the following are true:

ρ(g1g2) = ρ(g1)ρ(g2) ∀ g1, g2 ∈ G

ρ(g−1) = ρ(g)−1 ∀ g ∈ G

ρ(e) = e

The kernel of a homomorphism ρ : G → H is the inverseimage of the identity element of H via ρ.


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The kernel is a normal subgroupA subgroup N of G is termed normal in G (in symbolsN E G ) if for any g ∈ G , gNg−1 = N.

The kernel of any group homomorphism ρ : G → His a normal subgroup of G.

Proof.Suppose the kernel of ρ is a subset N of G . First, we showthat N is a subgroup:

I ρ(e) = e by the definition of homomorphism, so e ∈ N

I ρ(gh) = ρ(g)ρ(h), so if ρ(g) = e and ρ(h) = e, thenρ(gh) = e. Hence g ∈ N, h ∈ N =⇒ gh ∈ N.

I ρ(g−1) = (ρ(g))−1, so if ρ(g) = e, ρ(g−1) = e. Hence,g ∈ N =⇒ g−1 ∈ N.

To show that N is normal, observe that if h ∈ N, thenρ(h) = e, hence ρ(g)ρ(h)ρ(g−1) = ρ(g)ρ(g−1) = e for allg ∈ G . Thus, ρ(ghg−1) = e for all g ∈ G , and thusghg−1 ∈ N.


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Every normal subgroup is a kernel

Given a group G and a normal subgroup N E G , there is anatural homomorphism from G with quotient N. This is ahomomorphism to the quotient group G/N.

Conversely, any surjective homomorphism from G withkernel N is equivalent to the canonical mapping G → G/N.


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Every normal subgroup is a kernel

Given a group G and a normal subgroup N E G , there is anatural homomorphism from G with quotient N. This is ahomomorphism to the quotient group G/N.Conversely, any surjective homomorphism from G withkernel N is equivalent to the canonical mapping G → G/N.


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The improper and the trivial subgroup

For any group, there are two important extreme subgroups:

I The improper subgroup(defined), which is the wholegroup. A subgroup that is not improper is termed aproper subgroup(defined)

I The trivial subgroup(defined), which is the one-elementsubgroup comprising the trivial element

Both of these are normal subgroups:

I The improper subgroup is a normal subgroup and thequotient group is a one-element group.

I The trivial subgroup is a normal subgroup and thequotient map is an isomorphism (that is, the quotient isthe whole group)


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What is a simple group?

A simple group(defined) is a nontrivial group for which any ofthese equivalent conditions hold:

I It has no proper nontrivial normal subgroup

I Any homomorphism from it must be either trivial orinjective

I It has no quotients other than itself and the trivial group


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Simple groups as the “primes” of group theory

Prima facie, there is a strong parallel between simple groupsand the prime numbers.

In N (the set of natural numbers), every number has twospecial divisors:

I The improper divisor, which is the number itself. Anyother divisor is termed a proper divisor.

I The trivial divisor, which is the number 1. Any otherdivisor is termed a nontrivial divisor.

A natural number is termed prime if it has no propernontrivial divisor.Thus, if normal subgroups are the equivalent of divisors ingroup theory, simple groups are the equivalent of primenumbers.


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Prima facie, there is a strong parallel between simple groupsand the prime numbers.In N (the set of natural numbers), every number has twospecial divisors:



A natural number is termed prime if it has no propernontrivial divisor.

Thus, if normal subgroups are the equivalent of divisors ingroup theory, simple groups are the equivalent of primenumbers.


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Prima facie, there is a strong parallel between simple groupsand the prime numbers.In N (the set of natural numbers), every number has twospecial divisors:



A natural number is termed prime if it has no propernontrivial divisor.Thus, if normal subgroups are the equivalent of divisors ingroup theory, simple groups are the equivalent of primenumbers.


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Where the parallel fails

In the case of natural numbers, if d is a divisor of n, andd ′ = n, we can recover the value of n as the product dd ′.In the case of groups, if N E G , then the quotient groupG/N can be thought of as the quotient of G by N. However:

I This quotient is not a subgroup, far less a normalsubgroup, of G

I Knowing the subgroup N and the quotient G/N doesnot give us enough information to recover G .

Thus, unlike the number theory case, the roles of N andG/N cannot be interchanged.


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Partial salvation

There is a particular construction called direct product whichparallels multiplication of integers. Every finite group has beexpressed as a direct product of groups each of which cannotbe further decomposed as a direct product of nontrivialgroups.

It is also true that a simple group cannot be decomposed asa direct product of nontrivial.However, it is not true that any group that isdirect-product-indecomposable is simple. Thus, the study ofsimple groups is not good enough to understand all groups.


Vipul Naik

Basic definitions

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Partial salvation

There is a particular construction called direct product whichparallels multiplication of integers. Every finite group has beexpressed as a direct product of groups each of which cannotbe further decomposed as a direct product of nontrivialgroups.It is also true that a simple group cannot be decomposed asa direct product of nontrivial.

However, it is not true that any group that isdirect-product-indecomposable is simple. Thus, the study ofsimple groups is not good enough to understand all groups.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side


Partial salvation

There is a particular construction called direct product whichparallels multiplication of integers. Every finite group has beexpressed as a direct product of groups each of which cannotbe further decomposed as a direct product of nontrivialgroups.It is also true that a simple group cannot be decomposed asa direct product of nontrivial.However, it is not true that any group that isdirect-product-indecomposable is simple. Thus, the study ofsimple groups is not good enough to understand all groups.


Vipul Naik

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Vipul Naik

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Simple Abelian groups are cyclic of prime order

If a group G is Abelian, then ghg−1 = h for all h ∈ G .Thus, any subgroup of G is normal in G .

Thus, a simple Abelian group is a nontrivial group with noproper nontrivial subgroups.Hence, it must equal the cyclic subgroup generated by anyelement, so it is either Z or Z/nZ.But Z has proper nontrivial subgroups (mZ for any m > 1).Also, if n is composite, then mZ/nZ is a nontrivial subgroupof Z/nZ for any proper nontrivial divisor m of n. Thus theonly possibility for a simple Abelian group is a cyclic groupof prime order.The converse (that any cyclic group of prime order is simple)follows from Lagrange’s theorem.


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If a group G is Abelian, then ghg−1 = h for all h ∈ G .Thus, any subgroup of G is normal in G .Thus, a simple Abelian group is a nontrivial group with noproper nontrivial subgroups.

Hence, it must equal the cyclic subgroup generated by anyelement, so it is either Z or Z/nZ.But Z has proper nontrivial subgroups (mZ for any m > 1).Also, if n is composite, then mZ/nZ is a nontrivial subgroupof Z/nZ for any proper nontrivial divisor m of n. Thus theonly possibility for a simple Abelian group is a cyclic groupof prime order.The converse (that any cyclic group of prime order is simple)follows from Lagrange’s theorem.


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If a group G is Abelian, then ghg−1 = h for all h ∈ G .Thus, any subgroup of G is normal in G .Thus, a simple Abelian group is a nontrivial group with noproper nontrivial subgroups.Hence, it must equal the cyclic subgroup generated by anyelement, so it is either Z or Z/nZ.

But Z has proper nontrivial subgroups (mZ for any m > 1).Also, if n is composite, then mZ/nZ is a nontrivial subgroupof Z/nZ for any proper nontrivial divisor m of n. Thus theonly possibility for a simple Abelian group is a cyclic groupof prime order.The converse (that any cyclic group of prime order is simple)follows from Lagrange’s theorem.


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If a group G is Abelian, then ghg−1 = h for all h ∈ G .Thus, any subgroup of G is normal in G .Thus, a simple Abelian group is a nontrivial group with noproper nontrivial subgroups.Hence, it must equal the cyclic subgroup generated by anyelement, so it is either Z or Z/nZ.But Z has proper nontrivial subgroups (mZ for any m > 1).Also, if n is composite, then mZ/nZ is a nontrivial subgroupof Z/nZ for any proper nontrivial divisor m of n. Thus theonly possibility for a simple Abelian group is a cyclic groupof prime order.

The converse (that any cyclic group of prime order is simple)follows from Lagrange’s theorem.


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If a group G is Abelian, then ghg−1 = h for all h ∈ G .Thus, any subgroup of G is normal in G .Thus, a simple Abelian group is a nontrivial group with noproper nontrivial subgroups.Hence, it must equal the cyclic subgroup generated by anyelement, so it is either Z or Z/nZ.But Z has proper nontrivial subgroups (mZ for any m > 1).Also, if n is composite, then mZ/nZ is a nontrivial subgroupof Z/nZ for any proper nontrivial divisor m of n. Thus theonly possibility for a simple Abelian group is a cyclic groupof prime order.The converse (that any cyclic group of prime order is simple)follows from Lagrange’s theorem.


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Characteristic subgroups

Consider the following alternative definition of normality:An automorphism σ of a group G is termed inner(defined) ifthere is a h ∈ G , such that, for all g ∈ G , σ(g) = hgh−1. Asubgroup of a group is termed normal(defined) if it is invariantunder all inner automorphisms of the group. Note that anymap g 7→ hgh−1 is an automorphism.

A subgroup is said to be characteristic(defined) if it is invariantunder all automorphisms of the group. Clearly, anycharacteristic subgroup is normal.A group is said to be characteristically simple(defined) if ithas no proper nontrivial characteristic subgroup. Clearly anysimple group is characteristically simple.


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Consider the following alternative definition of normality:An automorphism σ of a group G is termed inner(defined) ifthere is a h ∈ G , such that, for all g ∈ G , σ(g) = hgh−1. Asubgroup of a group is termed normal(defined) if it is invariantunder all inner automorphisms of the group. Note that anymap g 7→ hgh−1 is an automorphism.A subgroup is said to be characteristic(defined) if it is invariantunder all automorphisms of the group. Clearly, anycharacteristic subgroup is normal.

A group is said to be characteristically simple(defined) if ithas no proper nontrivial characteristic subgroup. Clearly anysimple group is characteristically simple.


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Consider the following alternative definition of normality:An automorphism σ of a group G is termed inner(defined) ifthere is a h ∈ G , such that, for all g ∈ G , σ(g) = hgh−1. Asubgroup of a group is termed normal(defined) if it is invariantunder all inner automorphisms of the group. Note that anymap g 7→ hgh−1 is an automorphism.A subgroup is said to be characteristic(defined) if it is invariantunder all automorphisms of the group. Clearly, anycharacteristic subgroup is normal.A group is said to be characteristically simple(defined) if ithas no proper nontrivial characteristic subgroup. Clearly anysimple group is characteristically simple.


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Some subgroup-defining functions

A subgroup-defining function(defined) is a map that sendseach group to a subgroup of it, such that the choice ofsubgroup is preserved by group isomorphisms. That is, it is afunction f such that if σ : G → H is an isomorphism ofgroups, then σ sends f (G ) to f (H).

Examples of subgroup-defining functions:

1. The center(defined) is the set of elements that commutewith every element

2. The commutator subgroup(defined) is the subgroupgenerated by commutators (elements of the formx−1y−1xy)

3. The Frattini subgroup(defined) is the intersection of allmaximal subgroups.


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Some subgroup-defining functions

A subgroup-defining function(defined) is a map that sendseach group to a subgroup of it, such that the choice ofsubgroup is preserved by group isomorphisms. That is, it is afunction f such that if σ : G → H is an isomorphism ofgroups, then σ sends f (G ) to f (H).Examples of subgroup-defining functions:

1. The center(defined) is the set of elements that commutewith every element

2. The commutator subgroup(defined) is the subgroupgenerated by commutators (elements of the formx−1y−1xy)

3. The Frattini subgroup(defined) is the intersection of allmaximal subgroups.


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Subgroup-defining functions and simple groups

Any subgroup obtained as a result of a subgroup-definingfunction is invariant under isomorphisms, and hence, inparticular, it must be a characteristic subgroup. So, it mustbe a normal subgroup.

Thus, for a simple group, the result obtained by applyingeach subgroup-defining function is either the whole group orthe trivial subgroup.


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Subgroup-defining functions and simple groups

Any subgroup obtained as a result of a subgroup-definingfunction is invariant under isomorphisms, and hence, inparticular, it must be a characteristic subgroup. So, it mustbe a normal subgroup.Thus, for a simple group, the result obtained by applyingeach subgroup-defining function is either the whole group orthe trivial subgroup.


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For the center, commutator and Frattinisubgroup

The previous slide tells us that every finite simplenon-Abelian group is:

1. centerless(defined): The center cannot be the wholegroup, because that would make the group Abelian.Hence the center is trivial.

2. perfect(defined): The commutator subgroup cannot betrivial, because that would make the group Abelian.Hence, the commutator subgroup must be the wholegroup.

3. Frattini-free(defined): There exist maximal subgroups, sothe intersection of maximal subgroups cannot be thewhole group. Hence, the intersection must be trivial.

To summarize: any finite simple, non-Abelian group must becenterless, perfect and Frattini-free.


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Sequences of simple groups?

Are there naturally occurring sequences of groups all ofwhose members (except perhaps the first few) are simple? Ifthere are, how would we prove simplicity of all of them in asingle stroke?

More specifically if G1 ≤ G2 ≤ G3 . . . is an ascendingsequence of groups, how do we use the simplicity of Gn tohelp establish that Gn+1 is simple?One possible approach is to show the following two things:

1. Any nontrivial normal subgroup of Gn+1 must intersectGn nontrivially

2. There is no proper normal subgroup of Gn+1 containingGn.


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Are there naturally occurring sequences of groups all ofwhose members (except perhaps the first few) are simple? Ifthere are, how would we prove simplicity of all of them in asingle stroke?More specifically if G1 ≤ G2 ≤ G3 . . . is an ascendingsequence of groups, how do we use the simplicity of Gn tohelp establish that Gn+1 is simple?

One possible approach is to show the following two things:




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Are there naturally occurring sequences of groups all ofwhose members (except perhaps the first few) are simple? Ifthere are, how would we prove simplicity of all of them in asingle stroke?More specifically if G1 ≤ G2 ≤ G3 . . . is an ascendingsequence of groups, how do we use the simplicity of Gn tohelp establish that Gn+1 is simple?One possible approach is to show the following two things:




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The alternating groups

The approach outlined in the previous slides works for thealternating groups. Let An denote the alternating group on nelements. Then, think of the countable set N and considerAn as a subgroup of the group of permutations of N thatfixed all but the first n elements, and on the first n elements,is an even permutation. Clearly, A1 ≤ A2 ≤ A3 ≤ A4 ≤ . . .

Now, we show that for n ≥ 3, if N is a nontrivial normalsubgroup of An+1, N ∩ An is a nontrivial subgroup of An.


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The alternating groups

The approach outlined in the previous slides works for thealternating groups. Let An denote the alternating group on nelements. Then, think of the countable set N and considerAn as a subgroup of the group of permutations of N thatfixed all but the first n elements, and on the first n elements,is an even permutation. Clearly, A1 ≤ A2 ≤ A3 ≤ A4 ≤ . . .Now, we show that for n ≥ 3, if N is a nontrivial normalsubgroup of An+1, N ∩ An is a nontrivial subgroup of An.


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The alternating groups (continued)

Suppose N ∩ An is trivial. Then no nontrivial element of Nfixes the element (n + 1). For any m between 1 and n, let sbe an even permutation that takes (n + 1) to m (such an sexists for n ≥ 2). Since N = sNs−1 and N ∩ An is trivial,there is no nontrivial element of sNs−1 that fixes m. Theupshot is that no nontrivial element of N fixes any elementfrom 1 to n + 1, or equivalently, every nontrivial element ofN is a derangement.

Some arguments using cycle sizes, powers and conjugationcan now show that this is possible only if n = 3 where thegroup in question is the Klein-four group.


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Suppose N ∩ An is trivial. Then no nontrivial element of Nfixes the element (n + 1). For any m between 1 and n, let sbe an even permutation that takes (n + 1) to m (such an sexists for n ≥ 2). Since N = sNs−1 and N ∩ An is trivial,there is no nontrivial element of sNs−1 that fixes m. Theupshot is that no nontrivial element of N fixes any elementfrom 1 to n + 1, or equivalently, every nontrivial element ofN is a derangement.Some arguments using cycle sizes, powers and conjugationcan now show that this is possible only if n = 3 where thegroup in question is the Klein-four group.


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In the last slide, we showed that:For n ≥ 3, if N is a nontrivial normal subgroup of An+1,N ∩ An is a nontrivial normal subgroup of An.

Now we show that: the normal closure of An in An+1 is thewhole of An+1 when n ≥ 3.This follows from the fact that Am is generated by its3-cycles for all m ≥ 3, and that all 3-cycles are conjugate.


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In the last slide, we showed that:For n ≥ 3, if N is a nontrivial normal subgroup of An+1,N ∩ An is a nontrivial normal subgroup of An.Now we show that: the normal closure of An in An+1 is thewhole of An+1 when n ≥ 3.

This follows from the fact that Am is generated by its3-cycles for all m ≥ 3, and that all 3-cycles are conjugate.


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In the last slide, we showed that:For n ≥ 3, if N is a nontrivial normal subgroup of An+1,N ∩ An is a nontrivial normal subgroup of An.Now we show that: the normal closure of An in An+1 is thewhole of An+1 when n ≥ 3.This follows from the fact that Am is generated by its3-cycles for all m ≥ 3, and that all 3-cycles are conjugate.


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The classification theorem

The classification theorem states:Every finite simple non-Abelian group is either a member ofone of a collection of series (explicitly listed) or is one of the26 sporadic simple groups (also explicitly listed).


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The series part

In each of the series, there are finitely many exceptions tosimplicity (that is, some of the first few members may notbe simple):

1. Alternating groups An for n ≥ 5

2. Classical groups: Projective special linear group,projective unitary group, projective orthogonal group,projective symplectic group

3. An exceptional or twisted group of Lie type

I have already discussed the alternating groups. In this talk,I shall discuss a bit about the classical groups.


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General linear group

The group GLn(k), or the general linear group of order nover k, is the group of k-linear automorphisms of kn.Equivalently, it is the group of invertible n × n matrices overk under matrix multiplications.


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Proper nontrivial normal subgroups of GLn(k)GLn(k) has two important proper nontrivial normalsubgroups (and hence is not simple):

1. There is a natural surjective homomorphism:

det : GLn(k) → GL1(k)

that sends each linear transformation to its determinant.

The kernel of this homomorphism is a normal subgroupof GLn(k). When n ≥ 2 and k is not the field of twoelements, it is proper and nontrivial. It is termed thespecial linear group and is denoted as SLn(k).In fact, SLn(k) is the commutator subgroup of GLn(k).

2. Any scalar linear transformation commutes with everylinear transformation. In fact, the center is precisely thegroup of scalar invertible linear transformations. Thegroup of scalar invertible linear transformations is thus anormal subgroup of GLn(k).In fact, the group of scalar invertible lineartransformations is precisely the center of GLn(k).


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that sends each linear transformation to its determinant.The kernel of this homomorphism is a normal subgroupof GLn(k). When n ≥ 2 and k is not the field of twoelements, it is proper and nontrivial. It is termed thespecial linear group and is denoted as SLn(k).

In fact, SLn(k) is the commutator subgroup of GLn(k).2. Any scalar linear transformation commutes with every

linear transformation. In fact, the center is precisely thegroup of scalar invertible linear transformations. Thegroup of scalar invertible linear transformations is thus anormal subgroup of GLn(k).In fact, the group of scalar invertible lineartransformations is precisely the center of GLn(k).


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that sends each linear transformation to its determinant.The kernel of this homomorphism is a normal subgroupof GLn(k). When n ≥ 2 and k is not the field of twoelements, it is proper and nontrivial. It is termed thespecial linear group and is denoted as SLn(k).In fact, SLn(k) is the commutator subgroup of GLn(k).



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2. Any scalar linear transformation commutes with everylinear transformation. In fact, the center is precisely thegroup of scalar invertible linear transformations. Thegroup of scalar invertible linear transformations is thus anormal subgroup of GLn(k).

In fact, the group of scalar invertible lineartransformations is precisely the center of GLn(k).


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Projective special linear group

Let Z denote the subgroup of GLn(k) comprising invertiblescalars. Then:

1 E Z ∩ SLn(k) E SLn(k) E GLn(k)

The first quotient is the group µn of nth roots of unity in k,and third quotient is simply GL1(k). The second quotient istermed the projective special linear group and is denotedPSLn(k).It turns out that PSLn(k) is a simple group whenever k is afinite field and n ≥ 2 (except PSL2(F2)).


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Projective special groups

In general, given a subgroup H of the general linear group,there is the ascending sequence:

1 E Z ∩ SLn(k) ∩ H E SLn(k) ∩ H E H

The quotient SLn(k) ∩ H/(Z ∩ SLn(k) ∩ H) is termed theprojective special group corresponding to H.

Particular cases of H we are interested in: the orthogonalgroup, the symplectic group, the unitary group.


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Projective special groups

In general, given a subgroup H of the general linear group,there is the ascending sequence:

1 E Z ∩ SLn(k) ∩ H E SLn(k) ∩ H E H

The quotient SLn(k) ∩ H/(Z ∩ SLn(k) ∩ H) is termed theprojective special group corresponding to H.Particular cases of H we are interested in: the orthogonalgroup, the symplectic group, the unitary group.


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Orthogonal group

The orthogonal group(defined) over k, denoted On(k) isdefined as the set of matrices A such that AAt is the identity.

Take H = On(k). Then H ∩ SLn(k) is termed the specialorthogonal group(defined) and is denoted as SOn(k). Thegroup Z ∩ H ∩ SLn(k) is the same as the group Z ∩ SOn(k),and the quotient of SOn(k) by this group is termed theprojective special orthogonal group(defined) denoted asPSOn(k). It turns out that PSOn(k).PSOn(k) is not in general simple. We need to take thekernel by another normal subgroup (via the so-called spinornorm) to obtain a simple group.


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Orthogonal group

The orthogonal group(defined) over k, denoted On(k) isdefined as the set of matrices A such that AAt is the identity.Take H = On(k). Then H ∩ SLn(k) is termed the specialorthogonal group(defined) and is denoted as SOn(k). Thegroup Z ∩ H ∩ SLn(k) is the same as the group Z ∩ SOn(k),and the quotient of SOn(k) by this group is termed theprojective special orthogonal group(defined) denoted asPSOn(k). It turns out that PSOn(k).

PSOn(k) is not in general simple. We need to take thekernel by another normal subgroup (via the so-called spinornorm) to obtain a simple group.


Vipul Naik

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Two sides to it


The negative side


Orthogonal group

The orthogonal group(defined) over k, denoted On(k) isdefined as the set of matrices A such that AAt is the identity.Take H = On(k). Then H ∩ SLn(k) is termed the specialorthogonal group(defined) and is denoted as SOn(k). Thegroup Z ∩ H ∩ SLn(k) is the same as the group Z ∩ SOn(k),and the quotient of SOn(k) by this group is termed theprojective special orthogonal group(defined) denoted asPSOn(k). It turns out that PSOn(k).PSOn(k) is not in general simple. We need to take thekernel by another normal subgroup (via the so-called spinornorm) to obtain a simple group.


Vipul Naik

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Symplectic group

The symplectic group(defined) over k, denoted Spn(k), isdefined as the set of matrices A of order 2n such thatAPAt = P where P is the block matrix(

0 −II 0

)

It turns out that the symplectic group lies completely insideSL2n(k) (viz all symplectic matrices have determinant 1).Further, it is also true that Z ∩ Spn(k) = Z ∩ SL2n(k). Thequotient Spn(k)/(Z ∩ Spn(k) is termed the projectivesymplectic group(defined).All the projective symplectic groups are simple.


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Symplectic group


0 −II 0

)

It turns out that the symplectic group lies completely insideSL2n(k) (viz all symplectic matrices have determinant 1).Further, it is also true that Z ∩ Spn(k) = Z ∩ SL2n(k). Thequotient Spn(k)/(Z ∩ Spn(k) is termed the projectivesymplectic group(defined).

All the projective symplectic groups are simple.


Vipul Naik

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Symplectic group


0 −II 0

)

It turns out that the symplectic group lies completely insideSL2n(k) (viz all symplectic matrices have determinant 1).Further, it is also true that Z ∩ Spn(k) = Z ∩ SL2n(k). Thequotient Spn(k)/(Z ∩ Spn(k) is termed the projectivesymplectic group(defined).All the projective symplectic groups are simple.


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Projective special groups (continued)

The unitary group(defined) over C, denoted Un(C), is definedas the set of matrices A such that AA∗ = I where ∗ denotesthe conjugate-transpose operation. Here, conjugate meansthe entry-wise complex conjugate.

We can also define unitary group over a general field k, if wefix an involutive automorphism on that field that plays therole of conjugation. For instance, in a field of order p2, themap x 7→ xp is an involutive automorphism, and can thusplay the role of conjugation.Take H to be Un(k) for some field k with respect to aninvolutive automorphism. Then, H ∩ SLn(k) is denoted asSUn(k) and is termed the special unitary group(defined) overk. The quotient of this by its center is termed the projectivespecial unitary group(defined) and is denoted as PSUn(k).With one exception, projective special unitary groups arealways simple.


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The unitary group(defined) over C, denoted Un(C), is definedas the set of matrices A such that AA∗ = I where ∗ denotesthe conjugate-transpose operation. Here, conjugate meansthe entry-wise complex conjugate.We can also define unitary group over a general field k, if wefix an involutive automorphism on that field that plays therole of conjugation. For instance, in a field of order p2, themap x 7→ xp is an involutive automorphism, and can thusplay the role of conjugation.

Take H to be Un(k) for some field k with respect to aninvolutive automorphism. Then, H ∩ SLn(k) is denoted asSUn(k) and is termed the special unitary group(defined) overk. The quotient of this by its center is termed the projectivespecial unitary group(defined) and is denoted as PSUn(k).With one exception, projective special unitary groups arealways simple.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side



The unitary group(defined) over C, denoted Un(C), is definedas the set of matrices A such that AA∗ = I where ∗ denotesthe conjugate-transpose operation. Here, conjugate meansthe entry-wise complex conjugate.We can also define unitary group over a general field k, if wefix an involutive automorphism on that field that plays therole of conjugation. For instance, in a field of order p2, themap x 7→ xp is an involutive automorphism, and can thusplay the role of conjugation.Take H to be Un(k) for some field k with respect to aninvolutive automorphism. Then, H ∩ SLn(k) is denoted asSUn(k) and is termed the special unitary group(defined) overk. The quotient of this by its center is termed the projectivespecial unitary group(defined) and is denoted as PSUn(k).With one exception, projective special unitary groups arealways simple.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side


Outline







Vipul Naik

Basic definitions

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Two sides to it


The negative side


The constructive and the negative side

The constructive side of the classification is in preparing alist of simple groups (the series and the sporadic ones) andproving simplicity for all of them.

The negative side of the classification involves showing thatthere are no other finite simple groups.


Vipul Naik

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Two sides to it


The negative side


The constructive and the negative side

The constructive side of the classification is in preparing alist of simple groups (the series and the sporadic ones) andproving simplicity for all of them.The negative side of the classification involves showing thatthere are no other finite simple groups.


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Listing of the classical and exceptional groups

In the 1890s, the mathematician Sophus Lie undertook adetailed study of matrix groups in terms of their differentialstructure, thus starting a subject called “Lie theory”. Thisstudy was for matrix groups over fields like R and C (withan intrinsic differential structure). Building on work of Lie,Chevalley managed to characterize all infinite simple groupsin terms of matrix groups and their quotients by normalsubgroups.

The ideas used by Lie in the study of matrix groups over Rand C often extended formally to other structures such asfinite fields. In particular, exceptional groups of Lie typeconstructed over R and C constructed using Dynkindiagrams had analogues over finite fields, giving rise to finitesimple groups. These finite simple groups were dubbedexceptional groups of Lie type.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side


Listing of the classical and exceptional groups

In the 1890s, the mathematician Sophus Lie undertook adetailed study of matrix groups in terms of their differentialstructure, thus starting a subject called “Lie theory”. Thisstudy was for matrix groups over fields like R and C (withan intrinsic differential structure). Building on work of Lie,Chevalley managed to characterize all infinite simple groupsin terms of matrix groups and their quotients by normalsubgroups.The ideas used by Lie in the study of matrix groups over Rand C often extended formally to other structures such asfinite fields. In particular, exceptional groups of Lie typeconstructed over R and C constructed using Dynkindiagrams had analogues over finite fields, giving rise to finitesimple groups. These finite simple groups were dubbedexceptional groups of Lie type.


Vipul Naik

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Finding the sporadic simple groups

I The classification result would have been neater ifsporadic simple groups didn’t exist. So, Mathieu’sdiscovery of a simple group that just didn’t fit into anyof the series was a rude shock.

In the period 1860-1875, Mathieu discovered fivesporadic simple groups. Nobody could find any othersporadic simple groups.

I In 1950s, mathematicians decided to start work on aprogramme of classifying all the finite simple groups.The classification theorem they sought to prove wasthat: “every finite simple group is either in one of theseries (explicitly listed) or is among a finite list ofexceptions (of which some have been explicitly listed)”.The sporadic simple groups continued to be discoveredbetween 1960 and 1980, with the list expanding even asmathematicians were trying to show that it wascomplete.


Vipul Naik

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Two sides to it


The negative side



I The classification result would have been neater ifsporadic simple groups didn’t exist. So, Mathieu’sdiscovery of a simple group that just didn’t fit into anyof the series was a rude shock.In the period 1860-1875, Mathieu discovered fivesporadic simple groups. Nobody could find any othersporadic simple groups.



Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side




I In 1950s, mathematicians decided to start work on aprogramme of classifying all the finite simple groups.The classification theorem they sought to prove wasthat: “every finite simple group is either in one of theseries (explicitly listed) or is among a finite list ofexceptions (of which some have been explicitly listed)”.

The sporadic simple groups continued to be discoveredbetween 1960 and 1980, with the list expanding even asmathematicians were trying to show that it wascomplete.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side






Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side


The monster: the last of the sporadics

In 1973, Fischer and Greiss provided strong evidence for theexistence of a simple group, which they called the Monstergroup. In 1980, Greiss constructed the group by means of anexplicit representation.


Vipul Naik

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Simple groups: and where they’re foundSimple groups occur (for some strange reason) asautomorphisms, or symmetries, of symmetry-rich structures.This is true for the series of simple groups we saw:

I The alternating groups occur as symmetries of a setI The projective special linear groups occur as symmetries

of vector spacesI The projective special orthogonal, unitary and

symplectic groups occur as symmetries of vector spaceswith additional structure (a suitable bilinear form orinner product).

Similarly, the sporadic simple groups occur as symmetries ofsymmetry-rich one-off structures. For instance:

I The Higman-Sims group occurs as a subgroup of index2 in the automorphism group of the Higman-Sims graph

I The Monster group occurs as the automorphism groupof an algebra called the Greiss algebra

I The Conway groups occur as subquotients of theautomorphism group of the Leech lattice


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I The alternating groups occur as symmetries of a set

I The projective special linear groups occur as symmetriesof vector spaces

I The projective special orthogonal, unitary andsymplectic groups occur as symmetries of vector spaceswith additional structure (a suitable bilinear form orinner product).






Vipul Naik

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of vector spaces

I The projective special orthogonal, unitary andsymplectic groups occur as symmetries of vector spaceswith additional structure (a suitable bilinear form orinner product).






Vipul Naik

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of vector spacesI The projective special orthogonal, unitary and

symplectic groups occur as symmetries of vector spaceswith additional structure (a suitable bilinear form orinner product).






Vipul Naik

Basic definitions

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The negative side


Some groups can’t be simple

We observed that any simple non-Abelian group must becenterless, perfect and Frattini-free. Along similar lines, wecan put more restrictions on the nature and properties of asimple group, thus eliminating whole classes of groups ascandidates for simplicity.

The most promising route for this has so far been the Sylowtheory route. The Sylow theory route uses Sylow’s theoremon the existence and conjugacy of Sylow subgroups toimpose restrictions on the existence of groups with certainproperties. These manipulations eventually culminate inresults like:The odd-order theorem which implies that no finitenon-Abelian group of odd order can be simple.We shall explore more on this in the second talk.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side



We observed that any simple non-Abelian group must becenterless, perfect and Frattini-free. Along similar lines, wecan put more restrictions on the nature and properties of asimple group, thus eliminating whole classes of groups ascandidates for simplicity.The most promising route for this has so far been the Sylowtheory route. The Sylow theory route uses Sylow’s theoremon the existence and conjugacy of Sylow subgroups toimpose restrictions on the existence of groups with certainproperties. These manipulations eventually culminate inresults like:

The odd-order theorem which implies that no finitenon-Abelian group of odd order can be simple.We shall explore more on this in the second talk.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side



We observed that any simple non-Abelian group must becenterless, perfect and Frattini-free. Along similar lines, wecan put more restrictions on the nature and properties of asimple group, thus eliminating whole classes of groups ascandidates for simplicity.The most promising route for this has so far been the Sylowtheory route. The Sylow theory route uses Sylow’s theoremon the existence and conjugacy of Sylow subgroups toimpose restrictions on the existence of groups with certainproperties. These manipulations eventually culminate inresults like:The odd-order theorem which implies that no finitenon-Abelian group of odd order can be simple.

We shall explore more on this in the second talk.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side



We observed that any simple non-Abelian group must becenterless, perfect and Frattini-free. Along similar lines, wecan put more restrictions on the nature and properties of asimple group, thus eliminating whole classes of groups ascandidates for simplicity.The most promising route for this has so far been the Sylowtheory route. The Sylow theory route uses Sylow’s theoremon the existence and conjugacy of Sylow subgroups toimpose restrictions on the existence of groups with certainproperties. These manipulations eventually culminate inresults like:The odd-order theorem which implies that no finitenon-Abelian group of odd order can be simple.We shall explore more on this in the second talk.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side


Outline







Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side


From the abstract to the concrete

The classification theorem plays the same role as anyclassification: it takes us from a general concept to a veryspecific checklist. This means that now, to attack thequestion: “Does every simple group have property p?” itsuffices to check for all groups in this list. Thus, proofsabout finite simple groups can be reduced to a case-by-caseanalysis for individual groups.

This has been done, for instance, in the proof of the result:“In a finite simple group, every subgroup has alattice-theoretic complement”.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side


From the abstract to the concrete

The classification theorem plays the same role as anyclassification: it takes us from a general concept to a veryspecific checklist. This means that now, to attack thequestion: “Does every simple group have property p?” itsuffices to check for all groups in this list. Thus, proofsabout finite simple groups can be reduced to a case-by-caseanalysis for individual groups.This has been done, for instance, in the proof of the result:“In a finite simple group, every subgroup has alattice-theoretic complement”.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side


As a first step towards understanding groups

The ultimate aim is to classify all finite groups.

The classification of finite simple groups is a first stepbecause any finite group can be built in a certain way fromfinite simple groups. We shall explore this again in thesecond talk.


Vipul Naik

Basic definitions

Prerequisites









Two sides to it


The negative side


As a first step towards understanding groups

The ultimate aim is to classify all finite groups.The classification of finite simple groups is a first stepbecause any finite group can be built in a certain way fromfinite simple groups. We shall explore this again in thesecond talk.

simple groups and the classification problem (part...

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