Table Of ContentUNIVERSIDAD DE COSTA RICA ESCUELA DE MATEMATICA
T´opicos en la teor´ıa de grupos I Semestre 2014 5 horas, 5 cr´editos
Carta al estudiante
Nick Gill Oficina 422
2511 6559 [email protected]
Introduccio´n
La teor´ıa de grupos es el estudio formal de simetr´ıa en matem´aticas. Cada vez que un objeto matem´atico
exhibe alguna forma de simetr´ıa, un grupo est´a ‘presente’ en un cierto sentido.
En este curso nos centramos en los grupos finitos simples. En algu´n sentido general, todos los grupos
finitos se construyen a partir de estas estructuras particulares.
Objetivos generales
Uno de los logros m´as significativo en las matem´aticas modernas es la clasificaci´on completa de los grupos
finitos simples. Estudiaremos varias familias de grupos que aparecen en esta clasificac´ıon. Necesitaremos
definir estas familias, mostrar su simplicidad y, entonces, estudiaremos algunos de su propiedades estruc-
turales.
Objetivos especificos
Al final del curso el estudiante deber´a
(1) tener un entendimiento general del enunciado de la clasificaci´on de los grupos finitos simples;
(2) entenderelconceptodeunaacci´ondeungrupo,ypoderdescribiralgunaspropiedadesfundamentales;
(3) poder definir los grupos finitos alternantes, Alt(n), mostrar su simplicidad (para n ≥ 5), y describir
la estructura de sus subgrupos maximales;
(4) entender los conceptos de formas cuadr´aticas y sesquilineales, y poder describir algunas de sus
propiedades y sus tipos diversos;
(5) poder definir los grupos PSL (q), PSp (q) y PSU (q), y mostrar su simplicidad (para valores de n
n n n
y q apropiados);
(6) tener un primer entendimiento de los grupos ortogonales;
(7) (possiblemente) entender el enunciado del teorema de Aschbacher sobre los subgroups maximales de
grupos cl´asicos, y poder describir algunos de las clases de Aschbacher.
Contenidos
Los contenidos pueden cambiar pero, por el momento, esperamos estudiar lo siguiente:
(1) Una introducc´ıon general a los grupos finitos simples y su clasificaci´on;
(2) Un poco de la teor´ıa de categor´ıas;
(3) Una introducc´ıon a la teor´ıa de los grupos de permutaciones;
(4) Una primera mirada a los grupos alternantes, incluyendo una demonstrac´ıon de su simplicidad;
(5) El concepto de primitividad, y nociones relacionadas;
(6) Subgrupos m´ınimos normales y el socle;
(7) El teorema de O’Nan y Scott incluyendo (dependiendo de la preferencia de los estudiantes) una
demonstraci´on de la ‘versi´on d´ebil’;
(8) Series;
(9) Cuerpos y espacios vectoriales;
(10) Espacio proyectivo;
(11) Grupos lineales y su acci´on sobre el espacio proyectivo;
(12) Formas cuadr´aticas y sesquilineales;
(13) Isometr´ıas y el lema de Witt;
(14) Espacios polares;
(15) Los grupos simpl´ecticos;
(16) Los grupos unitarios;
1
2
(17) Los grupos ortogonales;
(18) El teorema de Aschbacher sobre los subgroups maximales de los grupos cl´asicos (este t´opico ser´a
incluido si tenemos tiempo, y si los estudiantes lo desean).
Evaluacio´n
Laevaluaci´onconsistir´adeex´amenesytareas. Serealizar´andosex´amenesparcialesenlasfechasindicadas
a continuaci´on.
Primer examen parcial 15 de Mayo, 10am
Segundo examen parcial 10 de Julio, 10am
La nota final se calcular´a ponderando el primer examen en un 40%, el segundo en un 40% y las tareas en
un 20%.
Horas de consulta
Las horas de consulta ser´an los lunes de las 2 a las 3 y los jueves de las 2 a las 4. Ser´an en la oficina 422.
Bibliografia
No hay un texto que incluya todo el material de este curso. Los textos principales son los siguientes:
• John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163,
Springer-Verlag, New York, 1996.
• Peter Cameron, Classical groups, notas de clase que se pueden encontrar en
http://maths.qmul.ac.uk/~pjc/class_gps
• Projective and polar spaces, notas de clase que se pueden encontrar en
http://maths.qmul.ac.uk/~pjc/pps/
(Estas constituyen la segunda edici´on. La primera edicı´on se public´o como QMW Maths Notes 13
en 1991.)
M´as textos de inter´es:
• Harald Simmons, An introduction to category theory. Notas de clase que se pueden encontrar en
http://www.cs.man.ac.uk/~hsimmons/zCATS.pdf
(He utilisado una pequen˜a parte de estas notas cuando he escrito el cap´ıtulo sobre la teor´ıa de las
categor´ıas.)
• Jean Dieudonn´e, La g´eom´etrie des groupes classiques. Un cla´sico, en franc´es.
• Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups. Este
libro contiene una demonstraci´on de (una versi´on refinada de) el teorema de Aschbacher sobre los
subgroups maximales de los grupos cl´asicos. Tambi´en contiene muchas informaciones sobre estos
grupos (y sobre los otros grupos finitos simples).
• DonaldTaylor,Thegeometryoftheclassicalgroups. Estetextoincluyetodoelmaterialdelasegunda
mitad del curso, y mucho m´as.
• Helmut Wielandt, Finite permutation groups. Otro cl´asico que da buena informaci´on de los temas
mayores dentro del desarollo de la teor´ıa de los grupos de permutaciones.
• Robert Wilson, Finite simple groups. Este texto se pueden encontrar en
http://link.springer.com/book/10.1007%2F978-1-84800-988-2
Notas de clases relacionadas se pueden encontrar en
http://www.maths.qmul.ac.uk/~raw/FSG/
• Joanna Fawcett, The O’Nan-Scott theorem for finite primitive permutation groups. Una tesis de
maestr´ıa muy bonita que da una demonstraci´on aut´onoma del teorema de O’Nan y Scott. Se puede
encontrar en
https://uwspace.uwaterloo.ca/bitstream/handle/10012/4534/Fawcett_Joanna.pdf
Tengo copias electr´onicas de la mayor parte de los textos dentro de estas listas, y puedo compartirlos si
lo desean.
FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS
NICK GILL
1. Introduction
Throughout this section G is a group. The group G is called simple if it is nontrivial and the only
normal subgroups of G are 1 and G.
{ }
(E1.1) Prove that if G is a finite simple abelian group, then G = C , the cyclic subgroup of
∼ p
order p, where p is a prime.
This course is motivated by a desire to understand the finite simple groups. As we shall see, when
we come to study series, an understanding of the finite simple groups takes us a long way towards
understanding all finite groups.
Oneofthegreatmathematicalachievementsofthelastcenturyhasbeenthecompleteclassification
ofthefinitesimplegroups. Thisclassification, theproofofwhichstretchesacrossthousandsofjournal
articles in work by dozens of authors, can be stated simply.
Theorem 1.1. (Classification of Finite Simple Groups) A finite simple group is isomorphic to one
of the following
(1) A cyclic group C , of order p where p is a prime;
p
(2) An alternating group Alt(n), where n 5;
≥
(3) A finite group of Lie type;
(4) One of 26 sporadic groups.
This course is, roughly speaking, split into two halves. In the first half we will study the second
type of simple group listed in CFSG, namely the alternating groups Alt(n). You have already met
these groups in an undergraduate course, but there are still many natural questions that one can ask
about them: What are their conjugacy classes? What are their automorphism groups? What are
their subgroups? We will give at least partial answers to all of these questions.
Our method in studying the alternating groups will be to exploit their natural structure as per-
mutation groups acting on sets with n elements. Thus we will spend quite a bit of time studying
permutation groups, which are objects of interest in their own right. 1
This start will set us up well for the second half of the course when we come to study the finite
classical groups. These are a subclass of the groups of Lie type, the others being known as the
exceptional groups of Lie type. Our analysis of the classical groups follows the original approach of
Jordan and, later, Dickson. In other words, we construct the classical groups as quotients of certain
subgroups of GL(V), the set of invertible linear transformations over a finite vector space V. These
subgroups have a natural action on the associated vector space V, and we can study this action using
permutation group theory in order to deduce properties of the relevant simple groups.
A brief note about what is missing: the two classes of finite simple group that we fail to discuss
are the exceptional groups of Lie type, and the 26 sporadic groups. The latter, at least, are a finite
set so we might argue that their omission is not so serious. On the other hand the sporadic groups
are among the most famous and beautiful objects in finite group theory, so their absence is regretted.
Unfortunately, it is their very sporadic-ness that makes them so hard to include – they do not submit
1Indeed it is worth noting that group theory first arose, via the work of Galois and his successors, as the study of
permutations of sets. In other words, in the beginning, permutation groups were the only objects studied from the
subject we now think of as group theory.
1
2 NICK GILL
easily to a uniform treatment and each sporadic group requires individual attention to be understood
properly. The keen student is encouraged to consult [Asc94].
The exceptional groups of Lie type are a different kettle of fish. They form an infinite class of
groups and, although they were discovered in a somewhat sporadic way through the first half of the
twentieth century, they now form part of a uniform theory of groups of Lie type that has its origins
with Chevalley, and later Steinberg, Ree and Tits. This uniform theory has the advantage that it
allows one to study all finite groups of Lie type (including the classical groups) in one fell swoop, but
it has the disadvantage (at least to my mind) of being somewhat more difficult than the approach
we shall take that pertains only to the classical groups.
In any case if one wishes to understand the classical groups properly, one should really understand
both approaches as each yields different insight.2 In this course we will not discuss the approach of
Chevalley, but we refer the interested reader to the beautiful book of Carter [Car89].
1.1. Prerequisites. I assume that you have done a basic course in group theory and are familiar
with the statements of the isomorphism theorems, Lagrange’s theorem, Sylow’s theorems and the
concept of a group action. I also assume that you have seen a definition of the sign of a permutation,
and have met the symmetric group, Sym(Ω), and the alternating group, Alt(Ω), for a set Ω.
1.2. Acknowledgments and sources. Writing this course has given me an excuse to read a great
deal of beautiful mathematical writing, for which I am very grateful.
IwanttorecordinparticulartheextensiveuseIhavemadeofunpublishedlecturenotesofJanSaxl
(Cambridge), Tim Penttila (UWA, now Colorado) and Michael Giudici (UWA), as well as published
work (or work available online) of Peter Cameron [Cama, Camb], Dixon and Mortimer [DM96],
Joanna Fawcett [Faw] and Harold Simmons [Sim].
The just-cited texts are all well worth reading. The keen student may also be interested in the
following:
(1) La g´eom´etrie des groupes classiques by Jean Dieudonn´e[Die63]. This is a classic, written in
French.
(2) The subgroup structure of the finite classical groups by Kleidman and Liebeck[KL90]. This
proves a refined version of Aschbacher’s theorem on the subgroup structure of the finite
classical groups. It also contains a wealth of other information on these groups (and other
almost simple groups).
(3) The geometry of the classical groups by Donald Taylor [Tay92]. This covers all the material
in the second half of this course plus a fair bit more.
(4) Finite permutation groups by Wielandt. Another classic which gives a good sense of the major
themes in the development of the theory of finite permutation groups.
2This is most clearly exhibited when one studies the subgroup structure of the classical groups. Subgroups that are
not almost simple are exhibited very clearly by the theorem of Aschbacher [Asc84] which uses the classical theory of
Jordan, whereas almost simple groups are often most clearly seen using the approach of Chevalley et al.
FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 3
2. A little category theory
A category C consists of
a class Obj of entities called objects.
•
a class Arw of entities called arrows.
•
two assignments source : Arw Obj and target : Arw Obj. These assignments are
• −→ −→
represented in the obvious way:
f
A B
−→
indicates that f is an arrow with source A and target B.
an assignment 1 : Obj Arw which, given an object A in C, yields an arrow 1 satisfying
A
• −→
A 1A A
−→
(In other words the source and target assignments of the distinguished arrow 1 are A itself.)
A
a partial composition Arw Arw Arw which has the following range of definition: Two
• × −→
arrows
f g
A B and B C
1 2
−→ −→
are composable, in that order, precisely when B and B are the same object. The resulting
1 2
arrow has form
fg
A C.
−→
In addition the category C must satisfy the following conditions:
(C1) Suppose we are given a diagram as follows:
f g h
A B C D.
−→ −→ −→
We require that (fg)h = f(gh). (In other words, composition is associative.)
(C2) Consider an arbitrary arrow f and the two compatible identity arrows, as follows:
A 1A A f B 1B B.
−→ −→ −→
We require that f1 = f = 1 f.
B A
Some notes:
We use words like ‘class’ and ‘assignment’ to allow for the possibility that Obj and Arw are
•
not sets. If they were sets (in which case C is called a small category), then ‘assignment’
would be the same as ‘function’.
When we write ‘fg’ for the composition of arrows f and g, we are simply fixing some notation
•
– do not confuse this with composition of functions (although for many categories, arrows
are indeed functions of a kind). You should also note our ordering which is somewhat uncon-
ventional, but which is chosen to be consistent with our later convention of studying groups
acting on the right.
A final piece of notation: given two objects A and B in C, we write Hom [A,B] for the class
C
•
of all arrows with source A and target B, and we call this the hom-class from A to B.
2.1. Examples of categories. We briefly discuss some examples of categories. The first type we
shall study – categories of structured sets – are far and away the easiest. In fact we will not use any
other type of category in our ensuing work, but it will be worth at least mentioning some other types
for the sake of our mathematical education.
4 NICK GILL
2.1.1. Structured sets. In these categories, each object is a ‘structured set’, i.e. a set equipped with
some extra gadgetry, and arrows are functions between the carrying sets which respect this gadgetry.
Rather than making such a notion precise3, let us list some examples:
Example 1.
Category Objects Arrows
Set sets functions
Pfn sets partial functions
Grp groups morphisms
AGrp abelian groups morphisms
Rng rings morphisms
Field fields morphisms
Pos posets monotone maps
Top topological spaces continuous maps
Vect vector spaces over a field K linear transformation
K
Mod R right R-modules over a ring R morphisms
−
R Mod left R-modules over a ring R morphisms
−
(E2.1) Prove that Set, Pfn, Grp, Top and Vect are categories.
K
D is a subcategory of a category C if the class of objects (resp. arrows) of D is a subset of the class
of objects (resp. arrows) of C and, moreover, D is a category. D is a full subcategory of a category
C if it is a subcategory and, moreover, if for all objects X,Y of D, Hom [X,Y] = Hom [X,Y].
D C
(E2.2) Which categories in Example 1 are (full) subcategories of some other category in
Example 1?
Another example of a structured-set category that is well-studied within permutation group theory
is the following:
Example 2. Our category is called SimpleGraph.
Objects: An object is a pair (V,E) where V is a set (the ‘vertices’) and E is a set
of subsets of V, each element of E having cardinality at most 2 (the ‘edges’).
Arrows: Consider an arrow
f
(V,E) (V ,E ).
� �
−→
Then f is just a function V V such that
�
→
e ,e E = f(e ),f(e ) E .
1 2 1 2 �
{ } ∈ ⇒ { } ∈
In combinatorics, f would be called a a graph morphism.
An easy variant of SimpleGraph is the category SimpleDigraph whose objects
are ‘directed graphs’. In this category objects are pairs (V,E) where V is a set and E
is a multiset of ordered pairs of elements of V. One defines arrows in the obvious way.
(E2.3) Complete the definition of SimpleDigraph and prove that it is a category.
(E2.4) Give the ‘right’ definition of the category Graph corresponding to graphs that are
not necessarily simple, i.e. which may have multiple edges between vertices.
The final such structured-set category we consider will turn out to be important in the second half
of the course when we study the classical groups.
Example 3. Let us begin with the category Vect defined above. We will study a
K
couple of variants of Vect :
K
Variant 1: More arrows
A semilinear transformation from V to W is a map T : V W such that
→
3Theprecisenotionisthatofaconcretecategory. Thisisacategoryequippedwithafaithfulfunctortothecategory
Set.
FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 5
(1) (v +v )T = v T +v T for all v ,v V;
1 2 1 2 1 2
∈
(2) there exists an automorphism α of k such that
(cv)T = cα(vT)
for all c k,v V.4
∈ ∈
Our new category is called VectS . The objects are vector spaces over K; arrows
K
are semilinear transformations. Clearly Vect is a subcategory of VectS .
K K
Variant 2: Dot product
Let us specify K = R (an analogous discussion holds for K = C). If V is a finite-
dimensional vector space over R, then V can be equipped with a Euclidean inner
product as follows: choose a basis b ,...,b for V and define the inner product of
1 n
{ }
two vectors x,y V to be
∈
n
x y := x y ,
i i
·
i=1
�
where x = x b and y = y b .5
i i i i
We will define three new categories. All have the same set of objects: these are
� �
pairs (V, ) where V is a finite dimensional vector space over R and is a Euclidean
· ·
inner product on V (in other words, objects are Euclidean spaces).
f
IVect : an arrow (V , ) (V , ) is a linear transformation f : V V such that, for
R 1 · −→ 2 · 1 → 2
all v,w V ,
1
∈
vf wf = v w.
· ·
Notethat the dotson each side ofthisequation represent differentinner products.
f
SVect : an arrow (V , ) (V , ) is a linear transformation f : V V for which
R 1 · −→ 2 · 1 → 2
there exists a c R such that for all v,w V1,
∈ ∈
vf wf = c(v w).
· ·
f
SSVect : an arrow (V , ) (V , ) is a semilinear transformation f : V V for which
R 1 · −→ 2 · 1 → 2
there exists a c R such that for all v,w V1,
∈ ∈
vf wf = c(v w).
· ·
The reason for the names of these categories will become clear when we come to the
study of isomorphisms.
(E2.5) Prove that VectS and IVect are categories.
K R
Our final example is not exactly a category of structured sets, but it has very much the same
flavour. It will be crucial in what follows.
Example 4. Our category is called G Set.
−
Objects: An object is a triple (G,Ω,φ) where G is a group, Ω is a set and φ is an
action of G on φ, i.e. φ is a map G Ω Ω satisfying the usual axioms.
× →
Arrows: An arrow (G,Ω,φ) (H,Γ,ψ) is a pair (α,β) where α : G H is a
−→ →
group morphism and β : Ω Γ is a total function. We require moreover that the
→
following diagram commutes:
4We will formally define the notion of ‘automorphism’ for a category C shortly; in particular a field automorphism
is an automorphism for the category Field. For now it may help to consider the example K = C and consider the
complex-conjugation map z z. This is a field automorphism of C.
→
5AbetterdefinitionofaEuclideaninnerproductisthatitisanon-trivialbilinearmapV V R. (Thisis‘better’
× →
because it does not involve a choice of basis.)
6 NICK GILL
φ
G Ω Ω
×
(α,β) β
ψ
H Γ Γ
×
(E2.6) Prove that G-Set is a category.
2.1.2. More exotic categories. The categories that we have met so far are all of a sort. Category
theory was developed not to deal with these, but to deal with categories that crop in far more exotic
ways. So one might consider, say, function categories, or categories of presheaves of a given category,
or categories of chain complexes, etc.
Rather than discuss the aforementioned exotic categories which are important for many reasons, I
will discuss an unimportant example that has the advantage of being easy to define and gives a tiny
flavour of what is possible.
Example 5. The objects are finite sets. An arrow
f
A B
−→
is a function
f : A B R.
× →
For each pair of arrows,
f g
A B C,
−→ −→
we define
fg : A C R, (a,c) f(a,b)g(b,c).
× → �→
b B
�∈
(E2.7) Prove that Example 5 yields a category.
2.2. Isomorphisms and automorphisms. A pair of arrows
f g
A B and B A
−→ −→
such that
fg = 1 and gf = 1
A B
form an inverse pair of isomorphisms. Each arrow is an isomorphism.
An arrow
f
A A
−→
(i.e. an arrow with source and target equal) is called an endormorphism. An arrow that is both an
endomorphism and an isomorphism is called an automorphism.
Given an object X, the set of all automorphisms of X is a group under composition. We call this
group the automorphism group of X and denote it Aut(X).
Consider the specific situation when X is an object in C, a category of structured sets. Thus X
can be thought of as a set (which we call Ω for now, to distinguish it from X), plus some extra
gadgetry, the ‘furnishings’ of the object. An automorphism of X will necessarily be a permutation
of the underlying set Ω. This means, in particular, that Aut(X) is group-isomorphic to a subgroup
of Sym(X), the symmetric group on X.
Example 6. In many cases Aut(X) is an object we have encountered before. For
example
FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 7
Category Aut(X)
Set Sym(X), the symmetric group on X
Grp Aut(X)
Top ???
Vect GL(X), the general linear group of X
K
(E2.8) What is Aut(X) when X is an object in Top?
f
In a category an arrow A B is called
−→
g
monic if, for each pair of arrows X A, we have
−→
• −h→
gf = hf = g = h;
⇒
g
epic if, for each pair of arrows B X, we have
−→
• −h→
fg = fh = g = h.
⇒
(E2.9) Show that
(1) an isomorphism is monic and epic;
(2) if C is a structured set (so that each arrow is carried by a total function between the
carriers of the two objects), then
injective = monic, and surjective = epic;
⇒ ⇒
(3) epic does not imply surjective in Ring;
(4) bijective does not imply isomorphism in Top.
Example 7. An isomorphism in G-Set is a permutation isomorphism. We will
discuss these in greater detail in due course.
Example 8. If X = (V,E) is an object in SimpleGraph, then Aut(X), the group
of automorphisms of the graph X, is the set of all bijective functions f : V V that
→
are arrows in SimpleGraph and whose inverse is an arrow in SimpleGraph.
(E2.10) What are automorphisms in Graph? Can you see why one needs a different defi-
nition in this context?
Example 9. Consider the variants on Vect which we defined earlier. For the first
K
objects V are vector spaces, for the remaining three, objects Rn are Euclidean spaces
(real vector spaces equipped with a Euclidean inner product).
(1) VectS : Aut(V) is the set of all invertible semilinear transformations of V, often
K
denoted ΓL(V).
(2) IVect : an arrow is an invertible linear transformation f : V V such that
R →
vf wf = v w for all v,w V. In other words f is an isometry of Rn, and
· · ∈
Aut(Rn) is the orthogonal group of Rn, denoted O(R,n) or, simply O(n) in the
literature.
(3) SVect : an arrow is an invertible linear transformation f : V V for which
R →
there exists c R such that vf wf = c(v w) for all v,w V. In other words f
∈ · · ∈
is a similarity of Rn, and Aut(Rn) is the similarity group of Rn.
(4) SSVect : an arrow is an invertible semilinear transformation f : V V for
R →
which there exists c R such that vf wf = c(v w) for all v,w V. In other
∈ · · ∈
words f is a semisimilarity of Rn, and Aut(Rn) is the semisimilarity group of
Rn.6
6This category is, in fact, the same as the previous, since R admits no automorphisms! Of course this construction
will also work for C or, indeed, any field you care to mention... And in these cases this category is interesting (as we
shall see).
8 NICK GILL
3. Group actions
Throughout this section G is a group and Ω is a set. A (right) action of G on Ω is a function
(1) ϕ : G Ω Ω, (g,ω) ωg
× → �→
such that
(A1) ω1 = ω for all ω Ω;
∈
(A2) (ωg)h = ωgh for all ω Ω and g,h G.
∈ ∈
We will refer to the triple (G,Ω,ϕ) as a G-set, since it is an object in the category G-Set. Let us
briefly discuss some examples, the first is particularly fundamental.
Example 10. Let Ω be a set and let G be any subgroup of Sym(Ω), the symmetric
group on Ω. The group G acts naturally on Ω via the action (1) where we write ωg to
mean the image of the element ω under the permutation g.
(E3.1) Verify that the function described in Example 10 is an action, i.e. that (A1) and
(A2) hold.
Example 11. Let V be a vector space and let G be any subgroup of GL(V), the
general linear group on V. The group G acts naturally on V via
G V V, (g,v) v g
× → �→ ·
Here we write v g to mean application (on the right) of the linear transformation g
·
to the vector v. If V is finite-dimensional, then we can take a basis and write v as a
row vector, g as a square matrix, and v g becomes just matrix multiplication.
·
(E3.2) Verify that the function described in Exercise 11 is an action.
(E3.3) Suppose that we changed the function described in Exercise 11 from v g to g v (so,
· ·
for instance, if V is finite-dimensional we consider v as a column vector and use matrix
multiplication). Show that this is not an action. Can you find a ‘natural’ adjustment to this
definition so that it becomes an action?
The next example is a specific instance of Example 10. To describe it we need a little bit of
notation. Suppose that A is a subset of a group G. Define
�A� := {a1a2···ak | k ∈ Z+,ai ∈ A or a−i 1 ∈ A}.
It should be clear that A is a group. In fact A is the smallest subgroup of G containing A and
� � � �
we refer to it as the group generated by A.
(E3.4) Give sufficient conditions such that
A = a1a2 ak k Z+,ai A .
� � { ··· | ∈ ∈ }
Give an example of a set A in a group G for which this inequality does not hold.
Example 12. Let Ω = 1,...,n with 3 n Z+. Consider the group G = g,h
{ } ≤ ∈ � � ≤
Sym(Ω) where
n n
g = (1,2,...,n) and h = (1,n 1)(2,n 2)... ,
− − �2� �2�
� �
The group G is known as D , the dihedral group of order 2n and in the ensuing
2n
exercise we will establish some standard facts about this group.
Let us make an observation about the action of G on the set Ω that will become
relevantshortly. WecanthinkofthesetΩasthesetofverticesofanobjectX = (Ω,E)
from the category SimpleGraph, where E is the set
1,2 , 2,3 ,..., n 1,n , n,1 .
{{ } { } { − } { }}
Clearly X can be represented by drawing a regular n-gon and labelling the vertices,
in order anti-clockwise, 1,...,n; see Figure 1 for an example when n = 5. Notice that
