- since the marks can only be .0 or .5, I have rounded each mark to its ``closest approximation'' (7.7 became 7.5, but 7.8 became 8.)

- the passing mark is 6- hence 5.5 is not enough (and since 5.5 is not allowed in the Osiris, you will see it there as a 5).

Final exam: Monday, January 28. Here are the exam exercises and here are the solutions .

Time: 14-17.

Place: Ruppertgebouw zaal.

Note: the book and and the notes taken during the lectures and the werkcollege's are allowed.

**FINAL MARKS:** the plan is that the marks will be made known in the first week of February, and we will
have a last meeting in which you can ask questions about the exam (and not only). Date and time will be comunicated
here as soon as possible.

- SEPTEMBER 25: **test 1**.

- OCTOBER 9: **test 2**.

- OCTOBER 29: **test 3**.

- DECEMBER 4 : **test 4**.

- JANUARY 15: **test 5 **.

**
- JANUARY 22: test 6 .
**

Op een mooie dinsdag middag zitten 18 studenten, waarvan geen twee de zelfde naam hebben, ijverig te werken aan de groepentheorie. De studenten zullen later op de middag door Puzzel-Piet bezocht worden, zodat ze samen een spel kunnen spelen. Voordat het spel begint, deelt Piet de regels al mee, zodat de studenten een strategie kunnen afspreken. Alle studenten spelen namelijk samen tegen Piet. De regels zijn als volgt: Piet maakt voor elke student een lootje met zijn of haar naam erop. Vervolgens nummert Piet de 18 lootjes achtereenvolgend van -3 tot en met 14. De studenten hebben geen enkele informatie over welke naam met welk nummer correspondeert. Piet wijst een willekeurige student aan die naar voren mag komen. Deze student kiest een nummer van een lootje en zwarte Piet laat zien welke naam er op het lootje met dan nummer staat. Vervolgens mag de student een ander lotnummer kiezen en zwarte Piet laat weer de bijbehorende naam zien. Dit gaat zo door totdat de student 9 nummers heeft gekozen en de 9 bijbehorende namen heeft gezien. Hierna verlaat de student de kamer. Dit gebeurt allemaal zonder dat de overige studenten enige informatie erbij krijgen over de gekozen nummers en bijbehorende namen. Nu is de tweede student aan de beurt, enz. tot en met de achttiende. Doel van het spel voor de studenten is, dat iedere student zijn eigen naam ziet op zo'n lootje bij Puzzel-Piet. In dat geval winnen alle studenten een lekkere stapel met pepernoten. Als tenminste 1 student niet zijn eigen naam te zien krijgt, dan wint Puzzel-Piet en krijgen de studenten de roe!

**Vragen**

1. Het is hopelijk duidelijk dat als alle studenten gewoon steeds 9 willekeurige lootjes kiezen, dat ze dan slechts een kans van 1/2 tot de 18e hebben om te winnen. Verzin een strategie waarbij de winkans zo groot mogelijk is (en bereken die kans).

2. Er bestaat een strategie waarbij de winkans voor de studenten groter dan 1/3 is! Bewijs dit.

Inzendingen dienen voor 5 december schriftelijk ingeleverd te worden bij de jury (bestaande uit Oliver, Sander en Tim). Er kunnen twee chocolade letters verdiend worden (een "S" en een "n"), bovenstaande jury beslist hierover.

De prijs is geen grapje,

Maar een belangrijk stapje.

De letters vormen een groep,

Neem het volgende onder de loep:

18=n is de eis,

Voor het winnen van de prijs!

Ik bekijk het lootje in mijn hand:

Volgens onze strategie,

Is mijn volgende keuze verwant,

met wat ik nu zie!

Zit je om het aantal permutaties van achttien getallen verlegen,

Met een cykel van lengte strikt groter dan negen,

Gebruik (en als mogelijk bewijs!) het volgende feit:

Het is kleiner dan twee derde keer achttien faculteit.

Uitwerkingen zijn tijdens het werkcollege op 4 december of eerder in te leveren. Oplossingen en prijzen worden een week later uitgereikt.

Here is some first info about the course:

The course is given every Monday, from 13:15 to 15:00, in MIN 211.

Starting date: September 10, 2007. Last lecture: January 18, 2008.

Lecturer: Marius Crainic.

Text book: "Groups and Symmetry" by M.A. Armstrong. Any other material which is not in the book will be posted on this page.

More general info about the course (content, examination, etc) can be found here.

The exercise classes will take place every Tuesday from 13:15 to 15:00. There are 3 groups for the werkcollege:

- group 1: will use room MIN 012 (36), and will work under the guidance of Aio Sander Dahmen and Maarten Dobbelaar.

- group 2: will use room MIN 016 (24), and will work under the guidance of Tim Baarslag.

- group 3: will use room MIN 204 (20), and will work under the guidance of Oliver Lorscheid.

Division into weeks (here is where I will add the part of the book that has been covered during each lecture, exercises, extra-material, etc):

**WEEK 37 (SEPTEMBER 10): [from Ch1, Ch2, Ch3 in the book]**: The definition of groups; remarks on the axioms (the uniqueness of the unit element and of the inverse), abelian groups, algebraic examples (complex numbers, real numbers, rational number etc with respect to usual multiplication and then with respect to the usual addition); the group $P$ of rotational symmetries of a regular pyramid on a 12-sided base (generated by $r$); the group $Z$ of rotational symmetries of a regular hexagon (generated by $r$ and $s$ and satisfying $rsr= s$).

**Exercises**: 2.3, 2.5, 2.8, 3.2, 3.3. **Extra-exercises**.

**WEEK 38 (SEPTEMBER 17):
[from Ch1, Ch4, Ch5, Ch7 in the book]**: Reminder about the def. of a group and examples P and Z, then the dihedral group Dn, then another group with 12 elements: T (rot. symm. of the regular tethraedron).

The group Zn of reminders modulo n. It is a group with respect to the addition modulo n. With respect to the multiplication modulo n however, even after removing the zero element, it may fail to be a group (example given: Z6). Remarked that Zn-{0} is a group iff n is a prime number with the promiss that it will have interesting consequences to "number theory".

Isomorphism of groups and examples.

The order of an element: definition + examples+ application (proving the fact that P, Z and T, although they have the same number of elements, theDe prijs is geen grapje, Maar een belangrijk stapje. De letters vormen een groep, Neem het volgende onder de loep: 18=n is de eis, Voor het winnen van het prijs! y are not isomorphic). By means of examples we remarked that, in finite groups, the order of each element divides the number of elements of the group. I mentioned this is true in (to be proven later).

The notion of a subgroup and the subgroup generated by an element. Remark that the number of elements of the subgroup generated by an element coincides with the order of the element. Then I mentioned the general result (to be proven later) which generalizes the provious one: in a finite group G, the number of elements of any subgroup of G must divide the number of elements of G. Next time we continue with examples of subgroups etc.

**Exercises**: 3.10, 4.2, 4.3, 4.5, 4.6, 4.7. **Extra-exercises**.

**WEEK 39 (SEPTEMBER 24): [from Ch. 5 and Ch. 11 in the book]** Fast reminder about the notion of group, isomorphisms of groups, the order ord(x) of an element, the notion of subgroup, the subgroup

Proved that the order of an element equals to the number of elements of the subgroup it generates, and that any cyclic group is isomorphic either to (Z, +) or to (Zn, +n) (of reminders modulo n) for some n.

Proved that any subgroup of (Z, +) is of type kZ for some integer k and then, more generally, that a subgroup of any cyclic group must itself be cyclic.

Inside D6 we found a subgroup isomorphic to D3 (also with the picture).

In all examples we have seen the striking fact on the possible number of elements of subgroups of a given group (and he possible values that the order of elements in a group can take). We proved, in general:

*Lagrange's Theorem*: If G is a finite group and H is a subgroup of G, then |H| must divide |G|.

Then we have derived the following corollaries:

Corollary L1: in a finite group G, the order of any element x must divide the number of elements of G.

Corollary L2: any group with p elements, where p is a prime number, must be isomorphich to (Zp, +p).

Corollary L3: In a finite group with n elements, the n-th power of any element of the group equals the identity.

Corollary L4 (*Fermat's Little's Theorem*): If p is a prime number and x is an integer which is not divisible by p, then the (p-1)-th power of x is congruent to 1 modulo p.

**Exercises**: 4.4, 4.8, 4.9, 5.1, 5.7, 5.11, 7.8, 7.10, 11.4. Extra exercises: no new ones (finish doing the ones left from last time).

**WEEK 40 (OCTOBER 1st):[from Ch. 5 and Ch. 6 in the book]** Fast reminder on the first lectures: groups, isomorphisms, subgroups, main examples, order of an element, subgroup generated by an element, cyclic groups, Lagrange's theorem and consequences. Then the precise definition for when a group is generated by certain elements.

**Permutations**: definition, the group S-index n, matrix-like representation of permutations and the rule for multiplication, the group of permutations of any set X, denoted S- index X.

**Example**: S-index 3 can be generated by 2 elements, e.g. by (12) and (13), or (12) and (23), or R:= (123) and S:= (12). The last two elements resemble the dihedral group D-index 3. Andm indeed, we see in the picture that S-index 3 is isomorphic to D-index 3. What about the group T of rotational symmetries of the tethraedron? We do not get all permutations in S-index 4, just half of them (came back to this later).

**Question**: what kind of ``special permutations'' can be used to generate S-index n? And how many do we really need?
The answer is in steps (sequence of theorems):

- we defined the notion of k-cycles, and expplained that they generate S-index n.

- we looked at transpositions (i.e. 2-cycles) which are quite simple (they have order 2) and there are in total n(n-1)/2 of them. We proved that they generate S-index n.

- then proved that the transpositions (1 2), (1 3), ..., (1 n) are enough to generate S-index n (hence n-1 transpositions are enough!). Then that (1 2), (2 3), ..., (n-1 n) are enough.

- finally, S-index n can be generated by 2 elements: the transposition (1 2) and the n-cycle (12 ... n).

Even/odd permutations: we defined the notion of even/odd permutations. Then we discussed the sign of a permutation and proved that the function sgn: S-index n ---> {-1, 1} has the property sgn(fg)= sgn(f)sgn(g).

**Exercises:** 6.2, 6.11, 6.3, 11.3, 11.4, 5.3, 5.10. **Extra-exercises**.

**WEEK 41 (OCTOBER 8)
[from Ch. 6 and Ch. 8 in the book]:** Explained a bijection between natural numbers and (positive) rational numbers. Reminder on permutations: k-cycles, transpositions, even/odd permutations, the main properties of the sign function

From the properties of the sign-function, we derived some corollaries:

- C1: a permutation f is even (or odd) if and only if sgn(f)= 1 (or sgn(f)= -1, respectively).

- C2: a permutation cannot be both even and odd at the same time.

- C3: the set of all even permutations in S_n form a subgroup of S_n, denoted A_n, and called the alternating group of order n. Moreover, |A_n|= n!/2.

As an example, we went back to the group of rotational symmetries of the regular tetrahedron, and showed that it is isomorphic to A_4. Also, for n greater or equal to 3, we showed that A_n is generated by 3-cycles.

Then we discussed:

** Cayley's theorem:** Any group G is isomorhich to a subgroup of the permutation group S_G.

Appart from the proof, we also. discussed the case of finite groups. First of all,

** Corollary to Cayley's theorem:** Any group G with n elements is isomorphic to a subgroup of S_n.

The **interesting question** for a finite group is to determine the SMALLEST possible natural number N such that G is isomorphic to S_N. Cayley's theorem says that such N's exist and it could be choosen to be |G|. In examples however, we can normally do much better. We saw this by looking at the rotational symmetry groups of the platonic solids.

**The tetrahedron:** we have a group with 12 elements, hence Cayley's theorem would only tell us that it is isomorphic to a subgroup of S_12, while the picture (discussed already) makes the connection (and the isomorphism) with S_4.

**The cube:** we get a group with 24 elements. Cayley's theorem gives us a subgroup of S_24. Looking at the picture, since the symmetries determine permutations of the vertices (8 of them!), we get a subgroup of S_8. We can do better by looking at the centers of the faces (6 of them!), to get a subgroup of S_6. The best however is obtained by looking at the diagonals of the cube (4 of them!) and remarking that any symmetry determines a permutation of the diagonals (and two different symmetries determine different permutations!), hence we get a subgroup of S_4. However, since |S_4|= 4!= 24= |G|, our group must be isomorphic to S_4.

**The octahedron:** we explained it is dual to the cube, hence has the same rotational symmetry group- isomorhic to S_4.

**The dodecahedron:** a similar discussion to the of the cube, 60 symmetries, and the key is to look at the five cubes inscribed in the octahedron. The resulting group is isomorphic to A_5.

**The icosahedron:** dual to the dodechaedron, hence we get A_5 again.

I will come back to the platonic solids next time.

** Exercises:** 8.1, 8.7. **Extra-exercises**.

** Bonus exercise:** (worth an extra 0.25 p on the final mark, provided it is delivered by the next Monday): find the relation between the sign function for permutations, and the determinant of matrices. Then, using that det(AB)= det(A) det(B) for any two matrices A and B, deduce that sgn(fg)= sgn(f)sgn(g) for any two permutations f and g.

**WEEK 42 (OCTOBER 15)
[from Ch. 10 in the book, and other parts]:** Reminder. Then formulated one of the main questions in group theory: Given n, find a complete list of groups with n elements (i.e. a collection of groups, G1, ..., Gk, each one of them having n elements, each two of them being non-isomorphic, and which is complete in the sense that any group with n elements is isomorphic to one in the list). We looked at some examples and started a table for values of n at most 12, describing lists of non-isomorphic groups with n elements, and aiming to get complete lists. In particular, we revisited some of the examples we have seen before (dihedral groups, symmetry groups, residues groups, etc). To fill in more groups, we needed a general construction which, out of two groups G and H, produces a new group GxH (the product of G and H).

Product of two groups: definition, then examples: (C, +) isomorphic to (R, +)x(R, +), but (C-{0}, *) is not isomorhic to (R-{0}, *)x (R-{0}, *) (where * means the usual multiplication of numbers). Instead, (C-{0}, *) is isomorphic to the product between R_{>0} and the unit circle. See also the exercises. Next, we looked at examples that can be helpful for compliting the lists of groups in our table. We proved that Z_2xZ_2 is not isomorphic to Z_4 (hence we get a new group with 4 elements), but Z_2xZ_3 is isomorphic to Z_6. More generally, we shouwed that Z_nxZ_m is cyclic if and only if (n, m)= 1 and, in this case, it is isomorphic to Z_nm.

The plan (for the first part of the next time) is to show that Z_4 and Z_2xZ_2 forms a complete list when n= 4, Z_6 and S_3 form a complete list when n= 6, and fill in more groups in our table (once we are done with this, we will move to matrix groups).

**WEEK 43 (OCTOBER 22)
[from several parts of the book]:** Reminder: a list of finite groups we have already seen, the product of two groups,
and afterwards we started filling in the lists of groups with a given number n of elements, for small values of n (between 2 and 12).
Then we proved that closed the list for n= 4 and n=6, hinting that, more generally, if n= 2p with p-a prime number, then
any group with 2p elements is isomorphic either to Z_{2p} or to D_p (bonus exercise). Then we looked at n= 8, where we had:

Z_8, Z_4x Z_2, Z_2x Z_2X Z_2, D_4

and we should that these are not isomorphic. I mentioned another group with 8 elements (the quaternion group Q-pp 70 in the book), and adding it to the list, the list becomes complete.

**WEEK 44 (OCTOBER 29)[Ch 9]:** This has mainly been some ``reminder'' on things about matrices, determinants and the space |R^n, which are not in the book (assumed to be known there). We discussed matrices, vectors in |R^n, the bijection between matrices and linear maps. Then we took the following properties of the determinant:

- det of the identity matrix is 1

- if we exchange two lines the determinant changes the sign

- writing a matrix as a column with entries vectors, the determinant is linear on each of the entries in the column.

and then we proved that any function from the set of n by n matrices to |R which has these properties must be the determinant, and gave an explicit formula involving a big sum over permutations (involving also the sign of permutations). Then we discussed the relevance of the determinant: to the solvability of systems of linear equations, for deciding when a set of n vectors in |R^n form a basis or not, and for deciding when a matrix is invertible.

Then we introduced GL_n(|R) (the general linear group), GL_n(|R)_{+}, O_n (the orthogonal group), SO_n(the special orthogonal group), GL_n(C), U_n (unitary group), SU_n (the special unitary group), pointing out that GL_n(|R) is isomorphic to the subgroup of all bijections of |R^n consisting of linear bijections. Also, given a matrix A, we mentioned that A is in O_n if and only if the associated linear map on |R^n preserves the inner product, or, equivalently, if it preserves the norm, or, equivalently, if it preserves distances.

Week 45 (November 5): NO LECTURE (tentamenweek)

**WEEK 46 (NOVEMBER 12)
[from Ch 9 and Ch 13 of the book]:** Reminder: relation between matrices and linear transformations, between
GL_n(R) and linear isomorphisms, between GL_n(R)^+ and orientation preserving linear isomorphisms, between
O(n) and isometries.
Then looked at particular cases (n= 1, 2, 3). Most interestingly, we have shown that

- any element of SO(2) is a rotation (in the plane), giving a group isomorphism between the unit circle and SO(2).

- any element of SO(3) comes from a rotation (in the space) around a line through the origin.

Interesting remark: given a solid in the space, the group of its rotational symmetries defines a subgroup of SO(3). From what we have seen in the previous lectures, we deduce that SO(3) has subgroups isomorphic to Z_n, D_n, or one of the rotational symmetry groups of the Platonic solids. Later in the book (Theorem 19.2) it is proven that any finite subgroup of SO(3) is isomorphic to one of these.

Finally, we did Cauchy's theorem: if G is a finite group, and p is a prime divizor of |G|, then G contains an element of order p.

**WEEK 47 (NOVEMBER 19)
[from Ch 15 of the book]:** Reminder: a general overview of what has been done in the course, and what will
come next. Also an informal discussion about braid groups, the (fundamental) groups associated to spaces, etc.

Then we we discussed left and right cosets and their properties, we defined G/H as the set of all left cosets of H in G, defined the index of H in G as the number of distinct left cosets of H in G (i.e. the number of elements of G/H) and we deduced that

|G|= |G/H| |H|

reproving the theorem of Lagrange (that |H| must divide |G|). Then we discussed the notion of normal subgroups, in a way slightly different from the book. We started by proving a theorem: for a subgroup H of G, the following are equivalent:

- xhx^{-1} belongs to H for all x in G, h in H.

- xHx^{-1}= H for all x in G.

xH= Hx for all x in G.

- the ``obvious operation'' on G/H:

(xH)* (yH)= (xy)H

is well-defined.

We then defined the notion of normal subgroup, as one which satisfies one of these equivalent conditions. The resulting group G/H is called the quotient of G by H (or modulo H).

Then we looked at examples:

**WEEK 48 (NOVEMBER 26)
[from Ch 15 and Ch. 16 of the book]:** Reminder: left cosets, quotients, normal subgroups, quotient groups, emphasized that, taking G/H means ``killing the elemets of H'', example: D_6/

Then discussed the commutator subgroup [G, G], theorem 15.6, with the example G= D_6 worked out.

Then discussed group homomorphisms, image and kernel, examples, remarked that a group homomorphism is injective if and onlyt if its kernel only contains the identity element, then proved the first isomorphism theorem with corollary: if f: G ---> G' is a surjective group homomorphism then Ker(f) is a normal subgroup of G and G/Ker(f) is isomorphic to G'. Then presented several examples.

Week 49 (December 3):

**WEEK 50 (DECEMBER 10)
[from Ch 17 of the book]:** Reminder: Group actions, orbits, set of orbits.
Then we looked at several examples (action of G on G by conjugation, action of a subgroup H on G,
the action that we used (without naming it) in the proof of the Cauchy theorem (action of Z_p), then
the one that we used to prove that the signature of the product of two permutations is
the product of the signatures (action of S_n on the set |R[X_1, ..., X_n] of polynomials
on n variables, action of Z_2 on R, then on R^2, then action of the circle S^1 on R^2, then the
matrix group GL_n, O(n), SO(n) acting on R^n.

Then, for a general action of a group G on a set X, we defined the stabilizer of an element x of X, denoted G_x, and proved that it is a subgrooup of G. Example: for the action of G on itself by conjugation, we get the centralizer of x (= all elements of G which commute with x). Then proved the Orbit-Stabilizer theorem and derived some consequences on the number of elements of $X$. Example: looked again at the proof of Cauchy theorem. Then stated two theorems: one saying that any group with p^2 elements (p-prime number) is isomorphic either to the cyclic group with p^2 elements or to Z_pxZ_p; the other one saying that in a group whose number of elements is a power of p, the center must be nontriovial. We proved the second one using the action of G on itself by conjugation.

Week 51 (December 17): NO LECTURE.

Week 52 (December 24): NO LECTURE (winter holidays).

Week 1 (December 31): NO LECTURE (winter holidays).

**WEEK 2 (JANUARY 7):[from Ch. 20 in the book]**
Reminder on actions of groups and the orbit-stabilizer theorem. Then we discussed the Sylow theorems. First we stated them and explained a bit the content. Then some examples, including the determination of all groups with 15 elements). Then the proofs.

**WEEK 3 (JANUARY 14):**
A very large part of the lecture was spent on reviewing some of the stuff done previously. First of all, there
was a short reminder on the notion of conjugation. Then a reminder on the results about finite groups, followed by
a discussion of groups with 2008 elements (we solved the exercise left last week- see the previous file). Then we
had a reminder about normal subgroups and quotient groups, giving two ways of thinking at a quotient group G/N: either
as the group of ``reminders modulo N'', or as ``the group obtained from G by killing N''. Then
we recalled the notion of generating set of a group, and defined the notion of ``freely generated''. Finaly, given
a set X, we constructed the free group generated by X, denoted F(X) (which consists of reduced words built out of the alphabet X).

**WEEK 4 (JANUARY 21):**
Reminder on generators, and free groups. Then we discussed presentation of groups
by generators and relations (the general construction, many examples, the
theorem that any group can be realized in this way). In the very last part, a colloring
problem (Chapter 18) was briefly discussed.

Week 5: TENTAMENWEEK.