Group Theory Textbooks: Recommended Books & Numerical Examples

[ElDavidas]

hi, can anybody recommend any good textbooks for group theory? I've just started a 3rd year course called groups and geometry and the lecture notes aren't the greatest of help.

The lecturer just writes down some algebra regarding the topics without much notes to support them and doesn't really explain well where things are coming from.

Ideally, I'd like a book showing lots of numerical examples with answers.

Thanks

 

[JasonRox]

hi, can anybody recommend any good textbooks for group theory? I've just started a 3rd year course called groups and geometry and the lecture notes aren't the greatest of help.

The lecturer just writes down some algebra regarding the topics without much notes to support them and doesn't really explain well where things are coming from.

Ideally, I'd like a book showing lots of numerical examples with answers.

Thanks

Check out Abstract Algebra by J. Gallian.

It is an applied look on Abstract Algebra.

 

[mathwonk]

you might try the classic, topics in algebra, by herstein. i recall it has lots of neat exercises about groups of various orders.

but there are two kinds of groups, abelian and non abelian, also finite and infinite.

if you stick to finite groups, then all abelian groups are direct sums of cyclic groups, so that's the whole story. It is kind of fun to decompose, for each n, the group of units of Z(n), into a direct sum of cyclic groups.

Note that when n is prime, that unit group is cyclic of order n-1.


non abelian ones are much more complicated.

the first interesting ones are "dihedral" groups, (symmetries of a polygon), then the platonic solid groups, symmetries of the cube, tetrahedron, and (the first really interesting one) the "icosahedral group", symmetries of the icosahedron, of order 60, and isomorphic to the alternating group A(5).


Of course one should be aware of the symmetric groups S(n), of all permutations of a set of n elements, in which A(n) is the unique normal subgroup of index 2.

I happen to like the Klein group of order 168, a finite matrix group which is also the automorphism group of the projective plane curve x^3Y + Y^3Z + Z^3X, (is that it? I forget after 30 years).

then there are the finite matrix groups, such as general or special linear groups over finite fields.

then once you look at various bilinear forms, there are matrix groups that preserve those form, such as orthogonal (matrices preserving length), or "symplectic" matrix groups (those preserving the basic alternating form).


then people who study riemann surfaces like to investigate the automorphism grioups of various riemann surfaces.

galois theorists are fond of galois groups of finite extensions of the rationals, but no one yet knows whether all finite groups occur this way.

it is interesting that at least all abelian finite groups do occur, and all nilpotent groups of odd order, but it seems controversial whether all finite solvable groups occur, much less all finite groups.


you might start by calvculating the group of rotations of a cube. e.g. how many elements does it have? hint: first compute the number of elements that leave one vertex fixed. then the total number of elements is 8 times that number. do you see why?

in fact this group seems to be isomorphic to the permutation group of 4 objects. proof? (and what are the 4 objects?)

 

[AKG]

hint: first compute the number of elements that leave one vertex fixed. then the total number of elements is 8 times that number. do you see why?

I don't see why. I mean I see that the number of rotations that leave one vertex fixed is 3 (and these rotations also leave the opposite vertex fixed, with one of them, identity, leaving every vertex fixed), and I know that the group has order 24, but I don't see why I can just look at the rotations that fix one corner and multiply by 8. I also see that there are 8 corners, but I don't see how this helps. No rotation fixes a single vertex, and there are rotations that fix no vertices.

However, the same thing seems to work with the tetrahedron. Given a vertex, there are 3 rotations that fix it, there are 4 vertices, and there are 3 x 4 = |A₄| elements of the group.

 

[mathwonk]

if there are 3 elements that leave your vertex where it is, then there are 3 elements that take it to each other possible destination as well. since a vertex may be sent to any other vertex by some rotation, there are 8 possible destinations for that vertex, and 3 thigns take it toe ach oie of them, so therer are 3(8) rotations in all.


this is the basic counting principle of group theory: given an action of a finite group on a set, pick any point x in that set. then the number of elements in the group equals the order of the orbit of that element, times the order of the isotropy subgroup of that point, i.e. the group leaving that point fixed.


this is used to study conjugacy of elements and subgroups, hence the nature of normal subgroups, etc...

i.e. there are various natural ways a group acts on itself, by translation for example. it also acts on its subgroups by translation and also by conjugation.


this formula then yields the fundamental formula for the index of the normalizer of an element, i.e. the index of the normalizer is the order of the conjugacy class, since the conjugacy class is the orbit of the eolement or subgroup under the action of conjugacy, and the normalizer is the subgroup leaving the object fixed, so the order of the group equals the order of the normalizer times the order of the conjnugacy class, hence the order of the group divided by the order of the normaliozer, i.e. the index of the nromalizer, equals the order of the conjugacy class, mumbo jumbo...


but it is all visible in the case of rotations of polyhedra.


so subgroups and conjugacy classes are just abstract versions of subgroups leaving a vertex fixed, and sets of vertices.

 

[AKG]

if there are 3 elements that leave your vertex where itm is, and there are 8 vertices then there are 3 elements that take it to each of the vertices. since there are 8 destinations for that vertex, and 3 elemenmhbst atke it to each destinåtion, there are in all 24 elements.

Ah, the Orbit-Stabilizer Theorem.

 

[mathwonk]

yes but this reveals that you may have merely memorized that theorem ratehr than understanding it.

at least at my school, grad students who have studied such stuff in groupo theory books, regularly fail to solve simple problems of this nature about actual concrete group actions.

 

[AKG]

I think the best way to learn anything in math (definitions, theorems, algorithms, etc.) is to first use them in very simple situations/problems. Then, use them in more complex problems, and the more the better. But it's always important, I find, to start with simple problems because doing that is what I think ensures understanding and prevents memorization. I probably never did that with the O-S theorem, but the example you gave is the kind of simple problem I probably needed. Thanks!

 

[mathwonk]

you are most welcome, and if you wish to pursue the topic along these liens visit my webpage and download my algebra course notes.

http://www.math.uga.edu/~roy/