[Undergraduate/Masters] Group Theory Exercises

[Breo]

Homework Statement

Exercises: https://mega.co.nz/#!YdIgjA7T!WmgIpFjCoO-elDyPtUkDNarm21sZ_xet6OTJndPGiRY
Text: https://mega.co.nz/#!pVRxVKIC!RfFZiW2atRNj9ycGa4Xx_7Nu5FO4a1e6wmyQVLCcGlQ

2. Homework Equations

The Attempt at a Solution

This is what I made, obviously all help would be appreciated :):

1) Given $$( \Re, +) $$ and $$( \Re, +)$$ a defined aplication $$\phi: \Re -> \Re \Longrightarrow \phi(g+g) = \phi(g) + \phi(g) = 2\phi(g); \forall g ( \Re, +). $$

Don't know how to end this...

2) $$ O(p, N-p)$$ Don't know what is the dimension.

$$SL(N,C)$$ " " ".

$$SU(N)$$ has dimendion $$N²-1 -> dim = 8.$$

$$U(N)$$ has dimension $$N² -> dim = 1.$$

3) Don't know how to work it out.

4) The transposition space. Order 3. Generator is identity.

I think this is totally wrong and also do not know how to end the exercise.

5) I do not know.

6) I am almost sure how to solve it but is kinda pain in the arse to post it here.

7) No idea. No idea how to solve the last 2 exercises.

 

[RUber]

You might want to take one problem at a time and type the question into the webpage. I don't use mega.com.

 

[Breo]

Statements

1. Find all homomorphisms from the group (R, +) to itself. Characterize
the isomorphisms. Can you give some subgroups? Are they normal?
Is so, give the corresponding quotient groups.

2. Give the dimension of the Lie groups O(3, 1), SL(2, C), SU(3) and U(1).

3. What is the center of the group GL(N, R). Is is a normal subgroup?

4. Consider a 2-dimensional crystal forming a triangular lattice. What
is the subgroup of translations leaving this lattice invariant? What
is the order of this group? What are its generators? What about the
subgroup of rotations?

5. Construct all groups of 4 elements G = {e, g, g , h} and write the group
table for them. Find their conjugacy classes.

6. Consider the set of all diagonal real N × N matrices. Is is a group?
What is its dimension? Is is a subgroup of GL(N, R)? Is it normal?

7. Is the set of reflexions in 3-dimensional space a group? What is its
dimension?Additional Exercises of Chapter I

• What are the conjugacy classes of the rotation group in 3 dimensions?