contains exactly one element. But such an orbit must have the form

g p = eG .

OrbH ((g, g, . . . , g)) ,

Hence g is the desired element of order p.

Later, we will meet the following type of action. Let k be a ¬eld and V a vector space over

k. Let GL| (V ) denote the group of all invertible k-linear transformations V ’’ V . Then for

any group G, a homomorphism of groups ρ : G ’’ GL| (V ) de¬nes a k-linear action of G on V

by

g · v = ρ(g)(v).

This is also called a k-representation of G in (or on) V . One extreme example is provided by

the case where G = GL| (V ) with ρ = IdGL| (V ) . We will be mainly interested in the situation

where G is ¬nite and k = R or k = C; however, other cases are important in Mathematics.

If we have actions of G on sets X and Y , a function • : X ’’ Y is called G-equivariant or

a G-map if

•(gx) = g•(x) (g ∈ G, x ∈ X).

An invertible G-map is called a G-equivalence (it is easily seen that the inverse map is itself

a a G-map). We say that two G-sets are G-equivalent if there is a G-equivalence between

them. Another way to understand these ideas is as follows. If Map(X, Y ) denotes the set of all

functions X ’’ Y , then we can de¬ne an action of G by

(g · •)(x) = g(•(g ’1 x)).

5. SOLVABLE GROUPS 17

Then the ¬xed point set of this action is

Map(X, Y )G = {• : g•(g ’1 x) = •(x) ∀x, g} = {• : •(gx) = g•(x) ∀x, g}.

So MapG (X, Y ) = Map(X, Y )G is just the set of all G-equivariant maps.

4. The Sylow theorems

The Sylow Theorems provide the beginnings of a systematic study of the structure of ¬nite

groups. For a ¬nite group G, they connect the factorisation of |G| into prime powers,

|G| = pr1 pr2 · · · prd ,

12 d

where 2 p1 < p2 < · · · < pd with pk prime, and rk > 0, to the existence of subgroups of

prime power order, often called p-subgroups. They also provide a sort of converse to Lagrange™s

Theorem.

Here are the three Sylow Theorems. Recall that a proper subgroup H < G is maximal if

it is contained in no larger proper subgroup; also a subgroup P G is a p-Sylow subgroup if

|P | = pk where pk+1 |G|.

Theorem 2.14 (Sylow™s 1st Theorem). A p-subgroup P G is maximal if and only if

it is a p-Sylow subgroup. Hence every p-subgroup is contained in a p-Sylow subgroup.

Theorem 2.15 (Sylow™s 2nd Theorem). Any two p-Sylow subgroups P, P G are

conjugate in G.

Theorem 2.16 (Sylow™s 3rd Theorem). Let P G be a p-Sylow subgroup with |P | = pk ,

so |G| = pk m where p m. Also let np be the number of distinct p-Sylow subgroups of G. Then

(i) np ≡ 1 (mod p);

(ii) m ≡ 0 (mod np ).

Finally, we end with an important result on chains of subgroups in a ¬nite p-group.

Theorem 2.17. Let P be a ¬nite p-group. Then there is a sequence of subgroups

{e} = P0 P1 ··· Pn = P,

with |Pk | = pk and Pk’1 Pk for 1 k n.

We also have the following which can be proved directly by the method in the proof of

Theorem 2.13.

Theorem 2.18. Let P be a non-trivial ¬nite p-group. Then the centre of P , Z(P ), is non-

trivial.

Sylow theory seemingly reduces the study of structure of a ¬nite group to the interaction

between the di¬erent Sylow subgroups as well as their internal structure. In reality, this is just

the beginning of a di¬cult subject, but the idea seems simple enough!

5. Solvable groups

Definition 2.19. A group G which has a sequence of subgroups

{e} = H0 H1 ··· Hn = G,

with Hk’1 Hk and Hk /Hk’1 cyclic of prime order, is called solvable (soluble or soluable).

18 2. RECOLLECTIONS AND REFORMULATIONS ON BASIC GROUP THEORY

Solvable groups are generalizations of p-groups in that every ¬nite p-group is solvable. A

¬nite solvable group G can be thought of as built up from the abelian subquotients Hk /Hk’1 .

Since ¬nite abelian groups are easily understood, the complexity is then in the way these sub-

quotients are ˜glued™ together.

More generally, for a group G, a series of subgroups G = G0 > G1 > · · · > Gr = {e} is

called a composition series for G if Gj+1 Gj for each j, and each successive quotient group

Gj /Gj+1 is simple. The quotient groups Gj /Gj+1 (and groups isomorphic to them) are called

the composition factors of the series, which is said to have length r. Every ¬nite group has

a composition series, with solvable groups being the ones with abelian subquotients. Thus, to

study a general ¬nite group requires that we analyse both ¬nite simple groups and also the ways

that they can be glued together to appear as subquotients for composition series.

6. Product and semi-direct product groups

Given two groups H, K, their product G = H — K is the set of ordered pairs

H — K = {(h, k) : h ∈ H, k ∈ K}

with multiplication (h1 , k1 ) · (h2 , k2 ) = (h1 h2 , k1 k2 ), identity eG = (eH , eK ) and inverses given

by (h, k)’1 = (h’1 , k ’1 ).

A group G is the semi-direct product G = N H of the subgroups N, H if N G, H G,

H © N = {e} and HN = N H = G. Thus, each element g ∈ G has a unique expression g = hn

where n ∈ N, h ∈ H. The multiplication is given in terms of such factorisations by

(h1 n1 )(h2 n2 ) = (h1 h2 )(h’1 n1 h2 n2 ),

2

’1

where h2 n1 h2 ∈ N by the normality of N .

An example of a semi-direct product is provided by the symmetric group on 3 letters, S3 .

Here we can take

N = {e, (123), (132)}, H = {e, (12)}.

H can also be one of the subgroups {e, (13)}, {e, (23)}.

7. Some useful groups

In this section we de¬ne various groups that are useful as test examples in the theory we will

develop. Some of these will be familiar although the notation may vary from that in previous

encounters with these groups.

7.1. The quaternion group. The quaternion group of order 8, Q8 , has as elements the

following 2 — 2 complex matrices:

±1, ±i, ±j, ±k,

where

10 i0 01 0i

1= = I2 , i= , j= , k= .

01 0 ’i ’1 0 i0

7.2. Dihedral groups.

Definition 2.20. The dihedral group of order 2n D2n is generated by two elements ±, β of

orders |±| = n and |β| = 2 which satisfy the relation

β±β = ±’1 .

The distinct elements of D2n are

±r , ±r β (r = 0, . . . , n ’ 1).

Notice that we also have ±r β = β±’r .

A useful geometric interpretation of D2n is provided by the following.

8. SOME USEFUL NUMBER THEORY 19

Proposition 2.21. The group D2n is isomorphic to the symmetry group of a regular n-

gon in the plane, with ± corresponding to a rotation through 2π/n about the centre and β

corresponding to the re¬‚ection in a line through a vertex and the centre.

7.3. Symmetric and alternating groups. The symmetric group on n objects Sn is best

handled using cycle notation. Thus, if σ ∈ Sn , then we express σ in terms of its disjoint cycles.

Here the cycle (i1 i2 . . . ik ) is the element which acts on the set n = {1, 2, . . . , n} by sending

ir to ir+1 (if r < k) and ik to i1 , while ¬xing the remaining elements of n; the length of this

cycle is k and we say that it is a k-cycle. Every permutation σ has a unique expression (apart

from order) as a composition of disjoint cycles, i.e., cycles with no common entries. We usually

supress the cycles of length 1, thus (123)(46)(5) = (123)(46).

It is also possible to express a permutation σ as a composition of 2-cycles; such a decompo-

sition is not unique, but the number of the 2-cycles taken modulo 2 (or equivalently, whether

this number is even or odd, i.e., its parity) is unique. The sign of σ is the number

sign σ = (’1)number of 2-cycles .

Theorem 2.22. The function sign : Sn ’’ {1, ’1} is a surjective group homomorphism.

The kernel of sign is called the alternating group An and its elements are called even per-

mutations, while elements of Sn not in An are called odd permutations. Notice that |An | =

|Sn |/2 = n!/2. Sn is the disjoint union of the two cosets eAn = An and „ An where „ ∈ Sn is

any odd permutation.

Here are the elements of A3 and S3 expressed in cycle notation.

A3 : e = (1)(2)(3), (123) = (13)(12), (132) = (12)(13).

S3 : e, (123), (132), (12)e = (12), (12)(123) = (1)(23), (12)(132) = (2)(13).

8. Some useful Number Theory

In the section we record some number theoretic results that are useful in studying ¬nite

groups. These should be familiar and no proofs are given. Details can be found in the book

[JBF] or any other basic book on abstract algebra.

Definition 2.23. Given two integers a, b, their highest common factor or greatest common

divisor is the highest positive common factor, and is written (a, b). It has the property that

any integer common divisor of a and b divides (a, b).

Definition 2.24. Two integers a, b are coprime if (a, b) = 1.

Theorem 2.25. Let a, b ∈ Z. Then there are integers r, s such that ra + sb = (a, b). In

particular, if a and b are coprime, then there are integers r, s such that ra + sb = 1.

More generally, if a1 , . . . , an are pairwise coprime, then there are integers r1 , . . . , rn such

that

r1 a1 + · · · + rn an = 1.

These are consequences of the Euclidean or Divison Algorithm for Z.

EA: Let a, b ∈ Z. Then there are unique q, r ∈ Z for which 0 r < |b| and a = qb + r.

It can be shown that in this situation, (a, b) = (b, r). This allows a determination of the

highest common factor of a and b by repeatedly using EA until the remainder r becomes 0,

when the previous remainder will be (a, b).

CHAPTER 3

Representations of ¬nite groups

1. Linear representations

In discussing representations, we will be mainly interested in the situations where k = R or

k = C. However, other cases are important and unless we speci¬cally state otherwise we will

usually assume that k is an arbitrary ¬eld of characteristic 0. For ¬elds of ¬nite characteristic

dividing the order of the group, Representation Theory becomes more subtle and the resulting

theory is called Modular Representation Theory. Another important property of the ¬eld k

required in many situations is that it is algebraically closed in the sense that every polynomial

over k has a root in k; this is true for C but not for R, however, the latter case is important in

many applications of the theory. Throughout this section, G will denote a ¬nite group.

A homomorphism of groups ρ : G ’’ GL| (V ) de¬nes a k-linear action of G on V by

g · v = ρg v = ρ(g)(v),

which we call a k-representation or k-linear representation of G in (or on) V . Sometimes V

together with ρ is called a G-module, although we will not use that terminology. The case where

ρ(g) = IdV is called the trivial representation in V . Notice that we have the following identities:

(Rep-1) (hg) · v = ρhg v = ρh —¦ ρg v = h · (g · v) (h, g ∈ G, v ∈ V ),

(Rep-2) g · (v1 + v2 ) = ρg (v1 + v2 ) = ρg v1 + ρg v2 = g · v1 + g · v2 (g ∈ G, vi ∈ V ),

(Rep-3) g · (tv) = ρg (tv) = tρg (v) = t(g · v) (g ∈ G, v ∈ V, t ∈ k).

A vector subspace W of V which is closed under the action of elements of G is called a G-

submodule or G-subspace; we sometimes say that W is stable under the action of G. It is usual

to view W as being a representation in its own right, using the ˜restriction™ ρ|W : G ’’ GL| (W )

de¬ned by

ρ|W (g)(w) = ρg (w).

The pair consisting of W and ρ|W is called a subrepresentation of the original representation.

Given a basis v = {v1 , . . . , vn } for V with dim| V = n, for each g ∈ G we have the associated

matrix of ρ(g) relative to v, [rij (g)] which is de¬ned by

n

(Rep-Mat) ρg vj = rkj (g)vk .

k=1

Example 3.1. Let ρ : G ’’ GL| (V ) where dim| V = 1. Then given any non-zero element

v ∈ V (which forms a basis for V ) we have for each g ∈ G a »g ∈ k satisfying g · v = »g v. By

Equation (Rep-1), for g, h ∈ G we have

»hg v = »h »g v,

and hence

»hg = »h »g .

From this it is easy to see that »g = 0. Thus there is a homomorphism Λ : G ’’ k— given by

Λ(g) = »g .

21

22 3. REPRESENTATIONS OF FINITE GROUPS

Although this appears to depend on the choice of v it is in fact independent of it (exercise). As

G is ¬nite, every element g ∈ G has a ¬nite order |g|, and hence we also have

»|g| = 1,

g

which we also leave as an exercise. This says that »g is a |g| th root of unity. Hence, given a

1-dimensional representation of a group, we can regard it as equivalent to such a homomorphism

G ’’ k— .

Example 3.2. As examples of the above, let us consider the following situations.

a) Take k = R. Then the only roots of unity in R are ±1, hence we can assume that

for a 1-dimensional representation over R, Λ : G ’’ {1, ’1}, where the codomain is a

group under multiplication. An interesting example of this is provided by the sign of

permutations, sign : Sn ’’ {1, ’1}.

b) Now take k = C. Then for each n ∈ N we have n distinct nth roots of unity in C— . We

will denote the set of all nth roots of unity by µn , and the set of all roots of unity by

µ∞ = µn ,

n∈N

where we use the inclusions µm ⊆ µn whenever m|n. These are abelian groups under

multiplication.

Given a 1-dimensional representation over C, the function Λ can be viewed as a

homomorphism Λ : G ’’ µ∞ , or even Λ : G ’’ µ|G| by Lagrange™s Theorem.

For example, if G = C is cyclic of order N say, then we must have for any 1-

dimensional representation of C that Λ : C ’’ µN . Note that there are exactly N of

such homomorphisms.

Example 3.3. Let G be a simple group which is not abelian. Then given a 1-dimensional

representation ρ : G ’’ GL| (V ) of G, the associated homomorphism Λ : G ’’ µ|G| has abelian

image, hence ker Λ has to be bigger than {eG }. Since G has no proper normal subgroups, we

must have ker Λ = G. Hence, ρ(g) = IdV .

Indeed, for any representation ρ : G ’’ GL| (V ) we have ker ρ = G or ker ρ = {eG }. Hence,

either the representation is trivial or ρ is an injective homomorphism, which therefore embeds

G into GL| (V ). This severely restricts the smallest dimension of non-trivial representations of

non-abelian simple groups.

Example 3.4. Let G = {e, „ } ∼ Z/2 and let V be any representation over any ¬eld not of

=

characteristic 2. Then there are k-vector subspaces V+ , V’ of V for which V = V+ • V’ and

the action of G is given by

v if v ∈ V+ ,

„ ·v =

’v if v ∈ V’ .

Proof. De¬ne linear transformations µ+ , µ’ : V ’’ V , by

1 1

µ+ (v) = (v + „ · v) , µ’ (v) = (v ’ „ · v) .