Some Review Topics for the Final Exam

 

First, review your notes, the textbook sections covered, and the homework problems.

Finish off any homework problems that you have not done yet. Ask about any confusing topics or problems in class or at office hours.

 

Review all major theorems covered. Have examples for each. See which homework problems used them. When the proof is of reasonable length, do you know how to prove it?

 

Some particular areas to review:

 

I. Examples: understand the group structure (meaning: order, number of generators, orders of elements, abelian yes/no, subgroups) for the following groups:

 

1) infinite groups: Z+, R+, Rx, ZxZ, Q/Z, R/Z, GL2(R) (For the last one you only need to know the subgroups corresponding to isometries in chapter 5.)

            Are any of these isomorphic to another? Prove it.

 

2) finite groups: understand the same as above for:

Sn, An

Also understand: normal subgroups, class equation for:

Cyclic groups

Dihedral Groups

Quartenions

 

3) Symmetry groups of  geometric figures

 

II: Recall group homomorphisms, the automorphism group, modular arithmetic.

 

III. Normal subgroups. Quotients. Simple groups. Normalizer subgroup N(H).

 

IV  Products and direct products.

 

V How to count & determine symmetry groups of figures, like the Platonic Solids.

 

VI Group actions.

a)      general case: group G acting on a set S.

Definition, Class Equation, Counting Formula.

b)      special cases: G acting on G by conjugation, G acting on G by left multiplication, G acting on  cosets in G/H, G acting on subgroups by conjugation

c)      Permutation Representation: what does it mean? When is it injective?

 

V. Sylow Theorems, especially applications. There will be at least one problem that needs you to apply them.

 

 

 

 

Extra  Review Problems:

 

1. Prove or disprove:

a) If G is a group in which every proper subgroup is cyclic, then G is cyclic.

b) If G is a group in which every proper subgroup is normal, G is abelian.

c) If H and K are subgroups of G such that H intersects K trivially, then HK is a subgroup.

d) The center of a group is a normal subgroup.

e) The image of an element x in G under a group homomorphism f: GàG’ has the same order as x.

f) If G has a normal subgroup H then G is isomorphic to H x G/H.

 

2. Find all cyclic subgroups of C4 x C2.

3. List the elements of A4 and their orders.

4. Determine Aut (C10), Aut(Zx).

 

5. Let H and N be normal subgroups of a group G, with N a subset of H. Define f: G/N à G/H by f(xN)=xH for all cosets xN in G/N.

a)      Prove this is a well-defined surjective group homomorphism.

b)      Show that (G/N)/(H/N)≈G/H.

 

6. Show that there is no simple group of order 28.

7. Show there is no simple group of order 30.

8. List all non-isomorphic abelian groups of order 200.

9. Classify all groups of order 8.

10. What are all the possible class equations (for G acting on G by conjugation) for a group G of order 27?

11. What is the class equation for the action of D4=<x,y : x4=1, y2=1, xy=yx-1 } on the set of cosets D4/H, where H={1, x2}? What is the Counting Formula?

 

Problem 6.4.7(a): Let H be a subgroup of prime index p. What are the possible numbers of conjugate subgroups of H?

 

Problem 5.9.3

 

Raleigh, [Sir] Walter Alexander (1861-1922)
”In an examination those who do not wish to know ask questions of those who cannot tell.”  J