Finite Simple Groups Classical Groups Conclusion

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Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Finite simple groups

Maris Ozols University of Waterloo

December 2, 2009

1 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Introduction

2 Examples (boring)

I {1G} E G I G E G

Deﬁnition A nontrivial group G is called simple if its only normal subgroups are {1G} and G itself.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Basics

Deﬁnition A subgroup N of a group G is called normal (write N E G) if gHg−1 = H for every g ∈ G.

3 Deﬁnition A nontrivial group G is called simple if its only normal subgroups are {1G} and G itself.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Basics

Deﬁnition A subgroup N of a group G is called normal (write N E G) if gHg−1 = H for every g ∈ G. Examples (boring)

I {1G} E G I G E G

3 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Basics

Deﬁnition A subgroup N of a group G is called normal (write N E G) if gHg−1 = H for every g ∈ G. Examples (boring)

I {1G} E G I G E G

Deﬁnition A nontrivial group G is called simple if its only normal subgroups are {1G} and G itself.

3 Deﬁnition A composition series of a group G is a maximal normal series (meaning that we cannot adjoin extra terms to it). Note: All factors in a composition series are simple.

Theorem (Jordan-H¨older) Every two composition series of a group are equivalent, i.e., have the same length and the same (unordered) family of simple factors.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Decomposition Deﬁnition A normal series for a group G is a sequence

{1G} = G0 C G1 C ··· C Gn = G.

Factor groups Gi+1/Gi are called the factors of the series.

4 Note: All factors in a composition series are simple.

Theorem (Jordan-H¨older) Every two composition series of a group are equivalent, i.e., have the same length and the same (unordered) family of simple factors.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Decomposition Deﬁnition A normal series for a group G is a sequence

{1G} = G0 C G1 C ··· C Gn = G.

Factor groups Gi+1/Gi are called the factors of the series. Deﬁnition A composition series of a group G is a maximal normal series (meaning that we cannot adjoin extra terms to it).

4 Theorem (Jordan-H¨older) Every two composition series of a group are equivalent, i.e., have the same length and the same (unordered) family of simple factors.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Decomposition Deﬁnition A normal series for a group G is a sequence

{1G} = G0 C G1 C ··· C Gn = G.

Factor groups Gi+1/Gi are called the factors of the series. Deﬁnition A composition series of a group G is a maximal normal series (meaning that we cannot adjoin extra terms to it). Note: All factors in a composition series are simple.

4 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Decomposition Deﬁnition A normal series for a group G is a sequence

{1G} = G0 C G1 C ··· C Gn = G.

Theorem (Jordan-H¨older) Every two composition series of a group are equivalent, i.e., have the same length and the same (unordered) family of simple factors.

4 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

The classiﬁcation theorem

5 Introduction The classification theorem Finite simple groups Classical groups Conclusion The classification theorem Theorem (Classification of finite simple groups) The following is a complete list of finite simple groups: 1. cyclic groups of prime order 2. alternating groups of degree at least 5 3. simple groups of Lie type 4. sporadic simple groups

6 The proof is being reworked and the 2nd generation proof is expected to span only a dozen of volumes.

Introduction The classification theorem Finite simple groups Classical groups Conclusion The classification theorem Theorem (Classification of finite simple groups) The following is a complete list of finite simple groups: 1. cyclic groups of prime order 2. alternating groups of degree at least 5 3. simple groups of Lie type 4. sporadic simple groups

Some statistics

I Proof spreads across some 500 articles (mostly 1955–1983).

I More than 100 mathematicians among the authors.

I It is of the order of 10,000 pages long.

6 Introduction The classification theorem Finite simple groups Classical groups Conclusion The classification theorem Theorem (Classification of finite simple groups) The following is a complete list of finite simple groups: 1. cyclic groups of prime order 2. alternating groups of degree at least 5 3. simple groups of Lie type 4. sporadic simple groups

Some statistics

I Proof spreads across some 500 articles (mostly 1955–1983).

I More than 100 mathematicians among the authors.

I It is of the order of 10,000 pages long. The proof is being reworked and the 2nd generation proof is expected to span only a dozen of volumes. 6 Introduction The classification theorem Finite simple groups Classical groups Conclusion The classification theorem Theorem (Classification of finite simple groups) The following is a complete list of finite simple groups: 1. cyclic groups of prime order 2. alternating groups of degree at least 5 3. simple groups of Lie type 4. sporadic simple groups

Headlines

I Cartwright, M. “Ten Thousand Pages to Prove Simplicity.” New Scientist 109, 26-30, 1985.

I Cipra, B. “Are Group Theorists Simpleminded?” What’s Happening in the Mathematical Sciences, 1995-1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 82-99, 1996.

6 I Proceed by induction on the order of the simple group to be classiﬁed and consider a minimal counterexample, i.e., let G be a ﬁnite simple group of minimal order such that G/∈ K.

I Note that every proper subgroup H of G is a K-group, i.e., has the property that B E A ≤ H ⇒ A/B ∈ K.

Starting point

I Odd Order Theorem (Feit-Thompson) Groups of odd order are solvable (i.e., all factors in composition series are cyclic).

I Equivalently, every ﬁnite non-abelian simple group is of even order.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Proof Strategy

I Let K be the (conjectured) complete list of ﬁnite simple groups.

7 I Note that every proper subgroup H of G is a K-group, i.e., has the property that B E A ≤ H ⇒ A/B ∈ K.

Starting point

I Odd Order Theorem (Feit-Thompson) Groups of odd order are solvable (i.e., all factors in composition series are cyclic).

I Equivalently, every ﬁnite non-abelian simple group is of even order.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Proof Strategy

I Let K be the (conjectured) complete list of ﬁnite simple groups.

I Proceed by induction on the order of the simple group to be classiﬁed and consider a minimal counterexample, i.e., let G be a ﬁnite simple group of minimal order such that G/∈ K.

7 Starting point

I Odd Order Theorem (Feit-Thompson) Groups of odd order are solvable (i.e., all factors in composition series are cyclic).

I Equivalently, every ﬁnite non-abelian simple group is of even order.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Proof Strategy

I Let K be the (conjectured) complete list of ﬁnite simple groups.

I Proceed by induction on the order of the simple group to be classiﬁed and consider a minimal counterexample, i.e., let G be a ﬁnite simple group of minimal order such that G/∈ K.

I Note that every proper subgroup H of G is a K-group, i.e., has the property that B E A ≤ H ⇒ A/B ∈ K.

7 I Equivalently, every ﬁnite non-abelian simple group is of even order.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Proof Strategy

I Let K be the (conjectured) complete list of ﬁnite simple groups.

I Proceed by induction on the order of the simple group to be classiﬁed and consider a minimal counterexample, i.e., let G be a ﬁnite simple group of minimal order such that G/∈ K.

I Note that every proper subgroup H of G is a K-group, i.e., has the property that B E A ≤ H ⇒ A/B ∈ K.

Starting point

I Odd Order Theorem (Feit-Thompson) Groups of odd order are solvable (i.e., all factors in composition series are cyclic).

7 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Proof Strategy

I Let K be the (conjectured) complete list of ﬁnite simple groups.

I Proceed by induction on the order of the simple group to be classiﬁed and consider a minimal counterexample, i.e., let G be a ﬁnite simple group of minimal order such that G/∈ K.

I Note that every proper subgroup H of G is a K-group, i.e., has the property that B E A ≤ H ⇒ A/B ∈ K.

Starting point

I Odd Order Theorem (Feit-Thompson) Groups of odd order are solvable (i.e., all factors in composition series are cyclic).

I Equivalently, every ﬁnite non-abelian simple group is of even order. 7 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Finite simple groups

8 Alternating groups n! A = {σ ∈ S | sgn(σ) = 1} |A | = n n n 2

For n ≥ 5 An is simple (Galois, Jordan) and non-abelian.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Cyclic and alternating groups

Cyclic groups

Cn = Z/nZ |Cn| = n

Cp is simple whenever p is a prime (by Lagrange’s theorem). Cp are the only abelian ﬁnite simple groups.

9 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Cyclic and alternating groups

Cyclic groups

Cn = Z/nZ |Cn| = n

Cp is simple whenever p is a prime (by Lagrange’s theorem). Cp are the only abelian ﬁnite simple groups.

Alternating groups n! A = {σ ∈ S | sgn(σ) = 1} |A | = n n n 2

For n ≥ 5 An is simple (Galois, Jordan) and non-abelian.

9 I classical Lie groups (6):

I linear groups (1) I symplectic groups (1) I unitary groups (1) I orthogonal groups (3)

I exceptional and twisted groups of Lie type (10)

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Groups of Lie type

Chevalley and twisted Chevalley groups There are 16 inﬁnite families that can be grouped as follows:

10 I linear groups (1) I symplectic groups (1) I unitary groups (1) I orthogonal groups (3)

I exceptional and twisted groups of Lie type (10)

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Groups of Lie type

Chevalley and twisted Chevalley groups There are 16 inﬁnite families that can be grouped as follows: I classical Lie groups (6):

10 I exceptional and twisted groups of Lie type (10)

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Groups of Lie type

Chevalley and twisted Chevalley groups There are 16 inﬁnite families that can be grouped as follows: I classical Lie groups (6):

I linear groups (1) I symplectic groups (1) I unitary groups (1) I orthogonal groups (3)

10 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Groups of Lie type

Chevalley and twisted Chevalley groups There are 16 inﬁnite families that can be grouped as follows: I classical Lie groups (6):

I linear groups (1) I symplectic groups (1) I unitary groups (1) I orthogonal groups (3)

I exceptional and twisted groups of Lie type (10)

10 I Mathieu groups (5)

I groups related to the Leech lattice (7)

I groups related to the Monster group (8)

I other groups (6)

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Sporadic groups

Sporadic groups There are 26 sporadic groups that can be grouped as follows:

11 I groups related to the Leech lattice (7)

I groups related to the Monster group (8)

I other groups (6)

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Sporadic groups

Sporadic groups There are 26 sporadic groups that can be grouped as follows:

I Mathieu groups (5)

11 I groups related to the Monster group (8)

I other groups (6)

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Sporadic groups

Sporadic groups There are 26 sporadic groups that can be grouped as follows:

I Mathieu groups (5)

I groups related to the Leech lattice (7)

11 I other groups (6)

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Sporadic groups

Sporadic groups There are 26 sporadic groups that can be grouped as follows:

I Mathieu groups (5)

I groups related to the Leech lattice (7)

I groups related to the Monster group (8)

11 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Sporadic groups

Sporadic groups There are 26 sporadic groups that can be grouped as follows:

I Mathieu groups (5)

I groups related to the Leech lattice (7)

I groups related to the Monster group (8)

I other groups (6)

11 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Classical groups

12 Ln(q) := Mn×n(Fq) GLn(q) := {M ∈ Ln(q) | det M 6= 0}

SLn(q) := {M ∈ GLn(q) | det M = 1} × ∼ × Z GLn(q) := αIn | α ∈ Fq = Fq PGLn(q) := GLn(q)/Z GLn(q)

Deﬁnitions and examples

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Notation Dictionary Preﬁxes Sets of matrices G general L linear S special Sp symplectic P projective U unitary Z center O orthogonal

13 Ln(q) := Mn×n(Fq) GLn(q) := {M ∈ Ln(q) | det M 6= 0}

SLn(q) := {M ∈ GLn(q) | det M = 1} × ∼ × Z GLn(q) := αIn | α ∈ Fq = Fq PGLn(q) := GLn(q)/Z GLn(q)

Deﬁnitions and examples

13 GLn(q) := {M ∈ Ln(q) | det M 6= 0}

SLn(q) := {M ∈ GLn(q) | det M = 1} × ∼ × Z GLn(q) := αIn | α ∈ Fq = Fq PGLn(q) := GLn(q)/Z GLn(q)

Deﬁnitions and examples

Ln(q) := Mn×n(Fq)

13 SLn(q) := {M ∈ GLn(q) | det M = 1} × ∼ × Z GLn(q) := αIn | α ∈ Fq = Fq PGLn(q) := GLn(q)/Z GLn(q)

Deﬁnitions and examples

Ln(q) := Mn×n(Fq) GLn(q) := {M ∈ Ln(q) | det M 6= 0}

13 × ∼ × Z GLn(q) := αIn | α ∈ Fq = Fq PGLn(q) := GLn(q)/Z GLn(q)

Deﬁnitions and examples

Ln(q) := Mn×n(Fq) GLn(q) := {M ∈ Ln(q) | det M 6= 0}

SLn(q) := {M ∈ GLn(q) | det M = 1}

13 PGLn(q) := GLn(q)/Z GLn(q)

Deﬁnitions and examples

Ln(q) := Mn×n(Fq) GLn(q) := {M ∈ Ln(q) | det M 6= 0}

SLn(q) := {M ∈ GLn(q) | det M = 1} × ∼ × Z GLn(q) := αIn | α ∈ Fq = Fq

13 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Notation Dictionary Preﬁxes Sets of matrices G general L linear S special Sp symplectic P projective U unitary Z center O orthogonal

Deﬁnitions and examples

Ln(q) := Mn×n(Fq) GLn(q) := {M ∈ Ln(q) | det M 6= 0}

SLn(q) := {M ∈ GLn(q) | det M = 1} × ∼ × Z GLn(q) := αIn | α ∈ Fq = Fq PGLn(q) := GLn(q)/Z GLn(q)

13 Description

I Take a set of matrices, e.g., Ln(q).

I Note that GLn(q) ⊂ Ln(q) is a group.

I GLn(q) is not simple, since SLn(q) is the kernel of × det : GLn(q) → Fq , so SLn(q) C GLn(q). I SLn(q) is still not simple, since Z SLn(q) C SLn(q). I Consider PSLn(q) = SLn(q)/Z SLn(q) .

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Constructing simple matrix groups “Recipe”

Z SLn(q) C SLn(q) C GLn(q) ⊂ Ln(q) PSLn(q) = SLn(q)/Z SLn(q)

14 I Note that GLn(q) ⊂ Ln(q) is a group.

I GLn(q) is not simple, since SLn(q) is the kernel of × det : GLn(q) → Fq , so SLn(q) C GLn(q). I SLn(q) is still not simple, since Z SLn(q) C SLn(q). I Consider PSLn(q) = SLn(q)/Z SLn(q) .

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Constructing simple matrix groups “Recipe”

Z SLn(q) C SLn(q) C GLn(q) ⊂ Ln(q) PSLn(q) = SLn(q)/Z SLn(q)

Description

I Take a set of matrices, e.g., Ln(q).

14 I GLn(q) is not simple, since SLn(q) is the kernel of × det : GLn(q) → Fq , so SLn(q) C GLn(q). I SLn(q) is still not simple, since Z SLn(q) C SLn(q). I Consider PSLn(q) = SLn(q)/Z SLn(q) .

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Constructing simple matrix groups “Recipe”

Z SLn(q) C SLn(q) C GLn(q) ⊂ Ln(q) PSLn(q) = SLn(q)/Z SLn(q)

Description

I Take a set of matrices, e.g., Ln(q).

I Note that GLn(q) ⊂ Ln(q) is a group.

14 I SLn(q) is still not simple, since Z SLn(q) C SLn(q). I Consider PSLn(q) = SLn(q)/Z SLn(q) .

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Constructing simple matrix groups “Recipe”

Z SLn(q) C SLn(q) C GLn(q) ⊂ Ln(q) PSLn(q) = SLn(q)/Z SLn(q)

Description

I Take a set of matrices, e.g., Ln(q).

I Note that GLn(q) ⊂ Ln(q) is a group.

I GLn(q) is not simple, since SLn(q) is the kernel of × det : GLn(q) → Fq , so SLn(q) C GLn(q).

14 I Consider PSLn(q) = SLn(q)/Z SLn(q) .

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Constructing simple matrix groups “Recipe”

Z SLn(q) C SLn(q) C GLn(q) ⊂ Ln(q) PSLn(q) = SLn(q)/Z SLn(q)

Description

I Take a set of matrices, e.g., Ln(q).

I Note that GLn(q) ⊂ Ln(q) is a group.

I GLn(q) is not simple, since SLn(q) is the kernel of × det : GLn(q) → Fq , so SLn(q) C GLn(q). I SLn(q) is still not simple, since Z SLn(q) C SLn(q).

14 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Constructing simple matrix groups “Recipe”

Z SLn(q) C SLn(q) C GLn(q) ⊂ Ln(q) PSLn(q) = SLn(q)/Z SLn(q)

Description

I Take a set of matrices, e.g., Ln(q).

I Note that GLn(q) ⊂ Ln(q) is a group.

I GLn(q) is not simple, since SLn(q) is the kernel of × det : GLn(q) → Fq , so SLn(q) C GLn(q). I SLn(q) is still not simple, since Z SLn(q) C SLn(q). I Consider PSLn(q) = SLn(q)/Z SLn(q) .

14 Theorem (Jordan–Dickson)

PSLn(q) is simple, except for n = 2 and q = 2 or 3.

Question What is the order of PSLn(q)?

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Linear groups PSLn(q)

Deﬁnition The projective special linear group is PSLn(q) := SLn(q)/Z SLn(q)

15 Question What is the order of PSLn(q)?

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Linear groups PSLn(q)

Deﬁnition The projective special linear group is PSLn(q) := SLn(q)/Z SLn(q)

Theorem (Jordan–Dickson)

PSLn(q) is simple, except for n = 2 and q = 2 or 3.

15 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Linear groups PSLn(q)

Deﬁnition The projective special linear group is PSLn(q) := SLn(q)/Z SLn(q)

Theorem (Jordan–Dickson)

PSLn(q) is simple, except for n = 2 and q = 2 or 3.

Question What is the order of PSLn(q)?

15 n 1. There are q − 1 non-zero vectors to choose v1 from. n 2. |{α1v1 | α1 ∈ Fq}| = q, so there are q − q choices for v2. 2 n 2 3. |{α1v1 + α2v2 | α1, α2 ∈ Fq}| = q , so there are q − q choices for v2. 4. etc.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Order of GLn(q) Claim 1 n n n 2 n n−1 |GLn(q)| = (q − 1)(q − q)(q − q ) ... (q − q ) n Y = qn(n−1)/2 (qi − 1) i=1

Proof. n Let v1, . . . , vn ∈ Fq be the columns of a matrix from GLn(q):

16 n 2. |{α1v1 | α1 ∈ Fq}| = q, so there are q − q choices for v2. 2 n 2 3. |{α1v1 + α2v2 | α1, α2 ∈ Fq}| = q , so there are q − q choices for v2. 4. etc.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Order of GLn(q) Claim 1 n n n 2 n n−1 |GLn(q)| = (q − 1)(q − q)(q − q ) ... (q − q ) n Y = qn(n−1)/2 (qi − 1) i=1

Proof. n Let v1, . . . , vn ∈ Fq be the columns of a matrix from GLn(q): n 1. There are q − 1 non-zero vectors to choose v1 from.

16 2 n 2 3. |{α1v1 + α2v2 | α1, α2 ∈ Fq}| = q , so there are q − q choices for v2. 4. etc.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Order of GLn(q) Claim 1 n n n 2 n n−1 |GLn(q)| = (q − 1)(q − q)(q − q ) ... (q − q ) n Y = qn(n−1)/2 (qi − 1) i=1

Proof. n Let v1, . . . , vn ∈ Fq be the columns of a matrix from GLn(q): n 1. There are q − 1 non-zero vectors to choose v1 from. n 2. |{α1v1 | α1 ∈ Fq}| = q, so there are q − q choices for v2.

16 4. etc.

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Order of GLn(q) Claim 1 n n n 2 n n−1 |GLn(q)| = (q − 1)(q − q)(q − q ) ... (q − q ) n Y = qn(n−1)/2 (qi − 1) i=1

Proof. n Let v1, . . . , vn ∈ Fq be the columns of a matrix from GLn(q): n 1. There are q − 1 non-zero vectors to choose v1 from. n 2. |{α1v1 | α1 ∈ Fq}| = q, so there are q − q choices for v2. 2 n 2 3. |{α1v1 + α2v2 | α1, α2 ∈ Fq}| = q , so there are q − q choices for v2.

16 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Order of GLn(q) Claim 1 n n n 2 n n−1 |GLn(q)| = (q − 1)(q − q)(q − q ) ... (q − q ) n Y = qn(n−1)/2 (qi − 1) i=1

16 Claim 3

|PSLn(q)| = |SLn(q)| /d where d = gcd(q − 1, n)

Conclusion n qn(n−1)/2 Y |PSL (q)| = (qi − 1) n gcd(q − 1, n) i=2

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Order of PSLn(q)

Claim 2 × × |SLn(q)| = |GLn(q)| / Fq where Fq = q − 1

17 Conclusion n qn(n−1)/2 Y |PSL (q)| = (qi − 1) n gcd(q − 1, n) i=2

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Order of PSLn(q)

Claim 2 × × |SLn(q)| = |GLn(q)| / Fq where Fq = q − 1

Claim 3

|PSLn(q)| = |SLn(q)| /d where d = gcd(q − 1, n)

17 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Order of PSLn(q)

Claim 2 × × |SLn(q)| = |GLn(q)| / Fq where Fq = q − 1

Claim 3

|PSLn(q)| = |SLn(q)| /d where d = gcd(q − 1, n)

Conclusion n qn(n−1)/2 Y |PSL (q)| = (qi − 1) n gcd(q − 1, n) i=2

17 It turns out that Sp2m(q) ⊂ SL2m(q). Deﬁnition The projective symplectic group is PSp2m(q) := Sp2m(q)/Z Sp2m(q)

Order

2 m qm Y |PSp (q)| = (q2i − 1) 2m gcd(q − 1, 2) i=1

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Symplectic groups PSp2m(q) Deﬁnition Let J := 0 Im . The set of symplectic matrices is −Im 0

n T o Sp2m(q) := S ∈ L2m(q) | SJS = J

18 Deﬁnition The projective symplectic group is PSp2m(q) := Sp2m(q)/Z Sp2m(q)

Order

2 m qm Y |PSp (q)| = (q2i − 1) 2m gcd(q − 1, 2) i=1

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Symplectic groups PSp2m(q) Deﬁnition Let J := 0 Im . The set of symplectic matrices is −Im 0

n T o Sp2m(q) := S ∈ L2m(q) | SJS = J

It turns out that Sp2m(q) ⊂ SL2m(q).

18 Order

2 m qm Y |PSp (q)| = (q2i − 1) 2m gcd(q − 1, 2) i=1

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Symplectic groups PSp2m(q) Deﬁnition Let J := 0 Im . The set of symplectic matrices is −Im 0

n T o Sp2m(q) := S ∈ L2m(q) | SJS = J

It turns out that Sp2m(q) ⊂ SL2m(q). Deﬁnition The projective symplectic group is PSp2m(q) := Sp2m(q)/Z Sp2m(q)

18 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Symplectic groups PSp2m(q) Deﬁnition Let J := 0 Im . The set of symplectic matrices is −Im 0

n T o Sp2m(q) := S ∈ L2m(q) | SJS = J

It turns out that Sp2m(q) ⊂ SL2m(q). Deﬁnition The projective symplectic group is PSp2m(q) := Sp2m(q)/Z Sp2m(q)

Order

2 m qm Y |PSp (q)| = (q2i − 1) 2m gcd(q − 1, 2) i=1 18 Deﬁnition The projective special unitary group is

2 2 2 PSUn q := SUn q /Z SUn q

Order

n(n−1)/2 n 2 q Y i i PSUn q = q − (−1) gcd(q + 1, n) i=2

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

2 Unitary groups PSUn q Deﬁnition q q2 For x ∈ Fq2 deﬁne x¯ := x . Note that x¯ = x = x. The set of unitary matrices is

2 n 2 T o Un q := U ∈ Ln q | U¯ U = In

19 Order

n(n−1)/2 n 2 q Y i i PSUn q = q − (−1) gcd(q + 1, n) i=2

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

2 Unitary groups PSUn q Deﬁnition q q2 For x ∈ Fq2 deﬁne x¯ := x . Note that x¯ = x = x. The set of unitary matrices is

2 n 2 T o Un q := U ∈ Ln q | U¯ U = In

Deﬁnition The projective special unitary group is

2 2 2 PSUn q := SUn q /Z SUn q

19 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

2 Unitary groups PSUn q Deﬁnition q q2 For x ∈ Fq2 deﬁne x¯ := x . Note that x¯ = x = x. The set of unitary matrices is

2 n 2 T o Un q := U ∈ Ln q | U¯ U = In

Deﬁnition The projective special unitary group is

2 2 2 PSUn q := SUn q /Z SUn q

Order

n(n−1)/2 n 2 q Y i i PSUn q = q − (−1) gcd(q + 1, n) i=2 19 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Orthogonal groups

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20 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion

Conclusion

21 I The ﬁnite simple groups are (Classiﬁcation Theorem):

I cyclic groups of prime order I alternating groups of degree at least 5 I simple groups of Lie type I sporadic simple groups I The classical groups are

I linear groups PSLn(q) I symplectic groups PSp2m(q) 2 I unitary groups PSUn q I orthogonal groups

Thank you for your attention!

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Conclusion

I Every finite group has a “unique” decomposition into finite simple groups(Jordan-HölderTheorem).

22 I The classical groups are

I linear groups PSLn(q) I symplectic groups PSp2m(q) 2 I unitary groups PSUn q I orthogonal groups

Thank you for your attention!

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Conclusion

I Every finite group has a “unique” decomposition into finite simple groups(Jordan-HölderTheorem). I The finite simple groups are (Classification Theorem):

I cyclic groups of prime order I alternating groups of degree at least 5 I simple groups of Lie type I sporadic simple groups

22 Thank you for your attention!

Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Conclusion

I Every finite group has a “unique” decomposition into finite simple groups(Jordan-HölderTheorem). I The finite simple groups are (Classification Theorem):

I cyclic groups of prime order I alternating groups of degree at least 5 I simple groups of Lie type I sporadic simple groups I The classical groups are

I linear groups PSLn(q) I symplectic groups PSp2m(q) 2 I unitary groups PSUn q I orthogonal groups

22 Introduction The classiﬁcation theorem Finite simple groups Classical groups Conclusion Conclusion

I Every finite group has a “unique” decomposition into finite simple groups(Jordan-HölderTheorem). I The finite simple groups are (Classification Theorem):

I cyclic groups of prime order I alternating groups of degree at least 5 I simple groups of Lie type I sporadic simple groups I The classical groups are

I linear groups PSLn(q) I symplectic groups PSp2m(q) 2 I unitary groups PSUn q I orthogonal groups

Thank you for your attention!