Summary week 1: Introduction; Cauchys Theorem; SylowsTheorems

1. Introduction

1.1 Tools Groups, subgroups, cosets, Lagranges Theorem, normal sub-

groups, quotient groups, homomorphisms. Isomorphism Theorems.

Group actions and the corresponding permutation representation. The Orbit

Stabilizer Theorem. For details, see for example [NST], the early chapters.

Most of the time, we use left actions. That is, if G is a group and is a set

then an action of G on is a map

G , (g, ) 7 g()

such that for all g, h G and we have

g(h()) = (gh)(), e() =

(where e is the identity of G).

If so then is said to be a G-set.

If so then is the disjoint union of orbits, ie of transitive G-sets.

Right actions are defined similarly. It is common to let the symmetric group

Sn act on the right of {1, 2, . . . , n}.

1.2 Orbit stabilizer theorem A transitive G-set is G-isomorphic to

cos(G : H) := {xH : x G} where H G, with action

g(xH) := (gx)H

Then || = |G||H|

.

Let = cos(G : H). Then := H cos(G : H). The stabilizer of is

Stab() = {g G : g() = } = {g G : gH = H} = H

If = x() for some x G then Stab() = xHx1.

1

1.3 Important G-spaces in this course

(1) Coset spaces cos(G : H) where H G.

(2) = G with conjugation action. The orbit of a is the conjugacy

class

aG = {gag1 : g G}

(called C(a) in the Mods lecture course). The stabilizer of a is known as the

centralizer of a in G, that is Stab(a) = CG(a) = {g G : ga = ag}. Then

|aG| =|G|

|CG(a)|.

(3) = the set of subgroups of G, with conjugation action. The orbit of

A is

{gAg1 : g G}

The stabilizer of A is known as the normalizer of A in G, that is Stab(A) =

{NG(A) = {g G : gAg1 = A}. We have

|{gAg1 : g G} =|G|

|NG(A)|.

1.4 More tools

Given a group G and A,B G define AB := {ab : a A, b B}.

Lemma 1 If G is a group and A,B G then |AB| = |A||B|/|A B|.

Lemma 2 If G is a group and A,B G then AB G AB = BA.

In particular if AEG or B EG then AB G.

Cauchys Theorem Let G be a finite group and p a prime number.If p divides |G| then there is some g G of order p.

We discuss two proofs (one in detail, and for other a sketch).

See also [NST].

Lemma 3 Assume G is a group of order pa for a prime p, and is

some finite G-set. Let FixG() be the set of fixed points of G in . Then

|FixG()| ||(modp).

2

DEF A p-group where p is a prime number is a group in which the order

of every element is a power of p.

Corollary of Cauchys Theorem A finite group is a p-group if and only

if |G| is a power of p.

2. Sylow p-subgroups

DEF Let p be a prime number. A Sylow p-subgroup of a group G is a

maximal p-subgroup.

That is, a p-subgroup P of G is a Sylow p-subgroup if

P Q G and Q a p-group P = Q

Note that this is different from the definition given in some books. With this

definition, the existence is obvious.

2.1 Lemma Assume that P is a Sylow p-subgroup of a group H where

P EH. If X is any p-subgroup of H then X P .

Sylows Theorems Let G be a finite group and suppose that |G| = pamwhere p is prime and p does not divide m. Then

(1) all Sylow p-subgroups have order pa,

(2) any two Sylow p-subgroups are conjugate in G,

(3) the number of Sylow p-subgroups is of the form kp+ 1 and divides |G|.

Thus any p-subgroup is contained in a subgroup of order pa, the subgroups of

order pa form a single conjugacy class. If P is one of them then the number

of conjugates is |G : NG(P )| 1( mod p).

3

Summary week 2: Sylows Theorems: examples andapplications. The Jordan-Holder Theorem

3. Sylows Theorems: examples and applications

We consider G = S4, and G = A5. As well, we find the Sylow p-subgroups

of the group GL(n, p) for any n 1 (if time).

Application 1 Let G be a group of order 12. Then either G = A4, or G

has a normal subgroup of order 3.

Application 2 Let G be a group of order 56. Then either G has a normal

subgroup of order 7, or G has a normal subgroup of order 8.

4. The Jordan-Holder Theorem

DEF 4.1 A simple group is a non-trivial group X in which the only normal

subgroups are {1} and X.

Observation 4.2 Let G be a group, and KEG. By the Third Isomorphism

Theorem, G/K is simple K is a maximal proper normal subgroup of G.

DEF 4.3 A composition series (CS) in a group G is a finite sequence of

subgroups of G such that

K0 = G > K1 > . . . > Km1 > Km = {1}

and where Ki+1 is a maximal proper normal subgroup of Ki for 0 i < m.

The simple groups Ki/Ki+1 are the composition factors of the series.

Note If G is finite then G has a CS.

Note A normal subgroup of a normal subgroup of G need not be normal in

G.

4

Jordan-Holder Theorem Let G be a finite group. Then all CS ofG have the same composition factors (up to isomorphism) with the same

multiplicities.

By the Jordan-Holder Theorem we can speak of the composition factors of

G. There are Jordan-Holder Theorems for modules over rings, and other

algebraic systems, with similar proofs.

Observation 4.4 Let G be a finite group and HEG. Let X1, X2, . . . , Xr be

the composition factors of H and let Y1, Y2, . . . , Ys be the composition factors

of G/H. Then the composition factors of G are X1, X2, . . . , Xr, Y1, Y2, . . . , Ys.

Example Let X and Y be finite simple groups, and let G := X Y , to

find all CS of G.

5

Summary week 3: Finite simple groups I. Solublegroups I.

5. Finite simple groups

Brief discussion of the list of the finite simple groups.

Lemma 5.1 Let g An and let K := {x1gx : x S(n)}. Let a1, a2, . . . , ak

be the lengths of the cycles of g. If a1, a2, . . . , ak are all different and all

odd then K splits as a union of two conjugacy classes of size |K|/2 in An.

Otherwise K is a single conjugacy class in An.

Theorem 5.2 The group A5 is simple.

Theorem 5.3 If n 5 then An is simple.

Corollary 5.4 The group An is the only non-trivial proper normal subgroup

of Sn for n 5.

6. Soluble groups I

DEF 6.1 A finite group G is said to be soluble if all its composition factors

are of prime order. [Some authors say solvable].

Observation Let G be a finite group, then if H E G then G is soluble if

and only if both H and G/H are soluble.

Observation If G is a finite soluble group and H G then H is soluble.

Examples (1) Abelian groups are soluble.

(2) Dn is soluble for all n.

(3) Sn is soluble if and only if n 4.

(4) Finite p-groups are soluble.

6

DEF 6.2 For a group G and a, b G define their commutator by [a, b] :=

a1b1ab.

Define the commutator subgroup (or derived group) of G by

G := [G,G] := [a, b] : a, b G

Lemma 6.3 (1) If H is normal in G and G/H is abelian then G H.

(2) If G H G then H is normal in G and G/H is abelian.

Observation If H EG then H EG.

DEF 6.4 Let G be a group. Define G[0] := G and inductively

G[k+1] := (G[k]) (k 0)

Then

G = G[0] G[1] . . .

is the derived series for G.

Terminology A series G = G0 G1 . . . Gn = {1} is called normal if

Gi is normal in G for all i. It is called subnormal if Gi is normal in Gi1 for

i 1; then the groups Gi1/Gi are the sections or factors of the series.

Theorem 6.5 For a group G the following are equivalent:

(1) G[k] = {1} for some k 0;

(2) G has a normal series with abelian sections;

(3) G has a subnormal series with abelian sections.

Note This gives the general definition: The (finite or infinite) group G is

said to be soluble if G[k] = {1} for some k 0 The least such k N is the

derived length of G. A finite group is soluble in this sense if and only if all

its CFs are of prime order.

7

Summary week 4: Soluble groups II. Small simplegroups

We have defined the derived series last week. A general group G is said to

be soluble if G[k] = {1} for some k 0. When G is finite, this is true if and

only if all CFs of G have prime order.

Observations (i) If G[r] = G[r+1] for some r then the derived series becomes

stationary at G[r], that is G[r+k] = G[r] for all k 0.

(ii) If G is simple non-abelian then G = G.

Example The derived series of Sn.

DEF 7.1 A finite group G is an elementary abelian p-group for some prime

p if G is abelian, such that xp = 1 for all x G.

Lemma 7.2 An elementary abelian p-group is isomorphic to ZpZp . . .

Zp.

Theorem 7.3 Assume G is a finite soluble group, and M G is a minimal

normal subgroup. Then M is an elementary abelian p-group for some prime

p.

Theorem 7.4 Assume G is soluble, and H < G is a maximal subgroup of

G. Then |G : H| = pa for some prime p and some a 1.

8. Small simple groups

Theorem 8.1 Suppose that |G| = 2m where m is odd. Then G has a normal

subgroup of order m. In particular if m > 1 then G is not simple.

Theorem 8.2 Let G be a group such that |G| = pam, where p is prime,

a 1, m 2 and pa does not divide 12m!. Then G is not simple.

Examples (1) Groups of order pq where p, q are prime, cannot be simple.

8

(2) Groups of order 24, 36, 40, or 48, cannot be simple.

Theorem 8.3 If G is a finite group and |G| < 60 then G is soluble.

Theorem 8.4 If G is a finite group and |G| = 60 then either G is soluble

or G = A5.

Summary week 5: Semi-direct products. Characters

9. Semi-direct products.

DEF 9.1 Given groups A and B, an extension of A by B is a group G with

a normal subgroup K such that K = A and G/K = B.

These always exist, eg the direct product A B. Usually there are many

extensions.

DEF 9.2 Let G be a group, K E G. Then G is said to split over K if

there exists H G such that K H = {1} and KH = G.

For example, S4 splits over V4. But Z4 does not split over Z2.

Construction Given groups A and B and a homomorphism : B

Aut(A). Write ab for the image of a under b. Write ab for (a1)b.

Define a group with underlying set AB as follows.

G := AB

1G := (1A, 1B) G

(a1, b1)(a2, b2) := (a1(a2)b11 , b1b2)

(a, b)1 := (ab, b1).

This is the semi-direct product of A by B with action , often denoted

A B.

Theorem 9.3 This construction produces a group G which is a split extension

of A by B.

9

10: Introducing characters

Let G be a finite group. Characters are constructed from representations.

Recall (from B2a):

DEF 10.1 (1) A linear representation of G is a homomorphism

: G GL(V )

where V is a finite-dimensional vector space over C.

(2) A matrix representation of G is a homomorphism

: G GL(n,C).

Note (Linear representation in V + basis ) matrix representation.

10.2 Let 1 : G GL(V1) and 2 : G GL(V2) be representations of G.

Then 1, 2 are said to be equivalent if there is an invertible linear transfor-

mation : V1 V2 such that for all g G

2(g) = 1(g) 1.

DEF 10.3 Let : G GL(V ) (or : G GL(n,C)) be a linear, or

matrix representation of G. The character associated to is the function

: G C defined by

(g) := trace((g)) (g G).

The character does not depend on the choice of a basis. As well, equivalent

representations have the same character.

We say that is a character of G if = for some representation .

Important examples 10.4 (1) The 1-dimensional trivial representation,

and the trivial character 1 : G C, where 1(g) = 1 (for all g G).

(2) Permutation representations: Let be a G-space, then G also acts on

the vector space C with basis , namely for g G define a linear map on

10

C by

(g) := [ 7 g()]

Then : G GL(C) is a representation. Let be the corresponding

character. Then

(g) = |fix(g)|

where fix(g) is the set of fixed points of g on .

Special case: If = G where the action is left multiplication, then the

representation is called the regular representation. Its character is

reg(g) =

{

|G| g = e0 g 6= e

(3) If G Sn, the sign representation .

(4) Representations of G = D2n on a 2-dimensional space.

Theorem 10.5 The character associated with a G-module V has the

following properties:

(1) (1G) = dimV ;

(2) (g) = (x1gx) for all g, x G;

(3) (g1) = (g) for all g G.

Lemma 10.6 Let : G GL(n,C) be a matrix representation of a

finite group G. Then for each g G, the matrix (g) is diagonalizable, with

diagonal entries k-th roots of 1 if g has order k.

Summary week 6: Representations and characters II.Character tables.

Let G be a finite group.

Much of the following is review of some representation theory.

DEF 11.1 Recall that the group algebra CG has vector space with basis

G, and where the algebra multiplication is obtained by taking the group

multiplication and extending it linearly.

11

11.2 (a) Given a representation : G GL(V ), then V becomes a CG-

module, if one sets

g v := (g)(v)

(b) Conversely, if W is a CG-module then we have a representation : G

GL(W ) where

(g) := [w 7 g w]

Observation CG-modules representations of G.

11.3 Assume V,W are CG-modules. A G-module homomorphism is a linear

map : V W such that

(g v) = g (v) (v V, g G).

Then HomG(V,W ) (or HomCG(V,W )) is the vector space of all G-module

homomorphisms from V to W . An isomorphism is an invertible homomor-

phism.

The modules V,W are isomorphic if and only if the corresponding represen-

tations are equivalent.

DEF 11.4 A representation : G GL(V ) is irreducible the corre-

sponding CG-module V is irreducible [simple]. That is if it does not have

any submodule 6= {0} or V .

A finite-dimensional G-module has a composition series. The Jordan-Holder

theorem holds.

DEF 11.4 Let V be an irreducible G-module, then the character V is

said to be irreducible.

Maschkes Theorem Every finite-dimensional G-module can be expressed

as a direct sum of simple submodules.

Schurs Lemma If X, Y are simple [irreducible] G-mdoules then

HomCG(X, Y ) =

{

C X = Y0 X 6= Y

12

Corollary Assume G is abelian. Then any irreducible representations is

1-dimensional.

Theorem 11.5 Let C1, C2, . . . , Ck} be the conjugacy classes in G. Define

the conjugacy class sums by

[Ci] :=

xCi

x CG

Then [C1], [C2], . . . , [Ck] for a basis for Z(CG), so dimZ(CG) = k.

Wedderburns Theorem The group algebra CG is isomorphic to adirect product of matrix algebras:

CG =

r

i=1

Mdi(C)

Consequences 11.6 (1) The simple CG-modules are (up to isomorphism)

X1, X2, . . . , Xr where under the isomorphism in 11.8

Xi (0, 0, . . . ,Cdi , 0, . . .)

In particular dimXi = di.

(2) r = dimZ(G) = k = the number of conjugacy classes of G.

(3) |G| =k

i=1(dimXi)2.

13

12. Character tables

Notation Let i be the character of Xi. These are the irreducible char-

acters of G. Label so that X1 is the trivial module, ie 1 is the trivial

character.

Let g1, g2, . . . , gk be conjugacy class representatives with g1 = e.

DEF 12.1 The character table of G is a k k array;

rows labelled by irreducible characters 1, 2, . . . kcolumns labelled by conjugacy class representatives g1, g2, . . . , gk;

entries = the values i(gj).

DEF 12.2 Define an inner product on the space of functions G C by

(, ) :=1

|G|

gG

(g)(g).

Theorem 12.3 Let 1, . . . , k be the irreducible characters of G. Then

(i, j) =

{

1 if i = j0 if i 6= j.

Corollary 12.4 The irreducible characters form an ON basis for the vector

space of class functions on G.

Corollary 12.5 Let V be a finite-dimensional G-module, let be its char-

acter, and let mi := (, i). Then mi Z0, and

V = m1X1 . . .mkXk

That is, a module is determined by its character.

Theorem 12.6 [Column Orthogonality ]

k

i=1

i(gr)i(gs) =

{

|CG(gr)| if r = s0 if r 6= s

14

Summary week 7: Linear characters, finding normalsubgroups, permutation characters, more character

tables.

An important fact (without proof)

Theorem 13.1 Let an irreducible character of G. Then the degree d =

(1) divides |G|.

DEF 13.2 A character is called linear if (e) = 1.

Linear characters are precisely the homomorphisms from G to C.

Lemma 13.3 If is a linear character and is a character of G then

is a character of G. It is irreducible is irreducible.

Corollary 13.4 The linear characters of G form an abelian group under

pointwise multiplication.

Remark If A is a finite abelian group then the group of all linear characters

of A is denoted A and called the dual group of A. One can show that if A

is a finite abelian group then A = A.

Lemma 13.5 Let K E G, and let H := G/K. For a character of H,

define : G C by

(g) := (Kg) (g G)

Then is a character of G. Moreover is irreducible is irreducible.

Corollary 1 Let k1 be the number of linear characters of G. Then k1 =

|G/G|.

15

Corollary 2 G is not simple there is some g G \ {e} and there is

some irreducible character i such that i(g) = i(e).

Lemma 13.6 Let : G GL(n,C) be a representation, and let be its

character, and K := ker(). Then

(1) |(g)| (e) for all g G;

(2) (g) = (e) g K;

(3) |(g)| = (e) g is a scalar matrix.

Corollary If is irreducible then |(g)| = (e) Kg Z(G/K).

14. Permutation characters

Let be a G-space. Recall

C is a CG-module, (permutation module). If is its character (permuta-

tion character) then

(g) = fix(g) (g G)

Let = 1 . . . t be the orbit decomposition. Define

ui :=

i

C, U := Sp{u1, . . . , ut} C.

Recall (Qu. 3 sheet 5) that U = {x C : xg = xg G}.

Corollary 14.1 Not Burnsides Lemma.

t = 1, =1

|G|

gG

fix(g)

DEF 14.2 Assume is a G-set. Define /G to be the set of G-orbits.

Lemma 14.3 Let G be a finite group, and let 1,2 be G-spaces. Then

1 , 2 = |(1 2)/G|.

16

DEF 14.4 Let G be a group, and a transitive G-space. The rank of G

in is |( )/G|. This is equal to |/G| for any .

If || 2 then rank(G) 2.

DEF 14.5 If rank(G) = 2 then G is said to be doubly transitive, or

2-transitive.

Thus G is doubly transitive G is transitive on and G is transitive on

\ {} for any .

Observation If G is 2-transitive on then = 1 + where is irre-

ducible.

[As well (g) Z for all g G.]

Examples

Summary: Tools for calculating character tables.

More examples.

Summary week 8

Three major theorems to be proved in week 8 (not examinable):

Theorem 15.1 Let an irreducible character of G. Then the degree d =

(1) divides |G|.

Theorem 16.2 [Burnside] Assume G is a finite group which has a conjugacy

class of size pr for p a prime number. Then G is not simple.

Theorem 16.3 [Burnsides paqb theorem ] Let p and q be distinct prime

numbers, and |G| = paqb. Then G is not simple.

17